# Prime number theorems for polynomials from homogeneous dynamics

Giorgos Kotsovolis  Department of Mathematics, Princeton University,
Princeton, NJ 08540<EMAIL_ADDRESS>and  Katharine Woo  Department of
Mathematics, Princeton University, Princeton, NJ 08540<EMAIL_ADDRESS>
###### Abstract.

We establish a new class of examples of the multivariate Bateman-Horn
conjecture by using tools from dynamics. These cases include the determinant
polynomial on the space of $n\times n$ matrices, the Pfaffian on the space of
skew-symmetric $2n\times 2n$ matrices, and the determinant polynomial on the
space of symmetric $n\times n$ matrices. In particular, let $(V,F)$ be any
pair among the following: $(\textrm{Mat}_{n},\det)$,
$(\textrm{Skew}_{2n},\textrm{Pff})$, and $(\textrm{Sym}_{n},\det).$ We then
obtain an asymptotic for

$\pi_{V,F}(T)=\\#\\{v\in V:\max(|v_{i}|)\leq T,F(v)\text{ is prime}\\},$

that matches the Bateman-Horn prediction.

The key ingredients of our proof are an asymptotic count for integral points
on the level sets of $F$ given by Linnik equidistribution, a geometric
approximation of the box by cones, and an upper bound sieve to bound the
number of prime values missed by the approximation. In the case of the
determinant polynomial on symmetric matrices, we must also use the Siegel mass
formula to compute the product of local densities for the main term.

Dedicated to Peter Sarnak
on the occasion of his 70th birthday

###### Contents

  1. 1 Introduction
  2. 2 General framework of invariant polynomials for homogeneous spaces
  3. 3 A prime number theorem on cones
  4. 4 $\epsilon$-cuttings
  5. 5 An upper bound sieve
  6. 6 Background on symmetric matrices and the Siegel Mass Formula
  7. 7 Symmetric matrices
  8. 8 Final remarks

##  1\. Introduction

Let $f(x)\in\mathbb{Z}[x]$ be an irreducible monic polynomial. It was
conjectured by Bunyakovsky in 1857 that $f(n)$ will be prime for infinitely
many $n\in\mathbb{N}$. This conjecture was made quantitative by Bateman and
Horn in [1], and is now known as the Bateman-Horn conjecture:

######  Conjecture 1.1 (The Bateman-Horn conjecture).

Let $f(x)$ be a nonconstant irreducible monic polynomial over $\mathbb{Q}$.
Then as $N\rightarrow\infty:$

$\\#\\{n\in[1,N]:f(n)\textrm{ is
prime}\\}\sim\mathfrak{S}\cdot\int_{0}^{N}\frac{dx}{\log^{+}(f(x))},$

where we write

$\log^{+}(y)=\begin{cases}\max(2,\log(y)),&y>0\\\ \infty,&y\leq 0,\end{cases}$

and where the singular series $\mathfrak{S}$ is defined as a product of local
densities:

$\mathfrak{S}=\prod_{p}\left(\frac{1-p^{-1}\\#\\{x\in\mathbb{F}_{p}:f(x)\equiv
0\bmod p\\}}{1-1/p}\right).$

While the Bateman-Horn conjecture has yet to be solved for any nonlinear
polynomial, mathematicians have made substantial progress proving prime number
theorems for multivariate polynomials. In the Appendix of [10], Destagnol and
Sofos wrote down the expected asymptotic for the multivariate Bateman-Horn.

######  Conjecture 1.2 (The Bateman-Horn conjecture for multivariate
homogeneous polynomials).

Let $F(\mathbf{x})$ be a nonconstant irreducible polynomial over $\mathbb{Q}$
in $n$ variables. For a box $\mathcal{B}\subset\mathbb{R}^{n}$, we denote
$T\mathcal{B}:=\\{t\mathbf{x}:t\in[0,T],\mathbf{x}\in\mathcal{B}\\}.$ Then, we
have that as $T\rightarrow\infty$,

$\\#\\{\mathbf{x}\in T\mathcal{B}:F(\mathbf{x})\textrm{ is
prime}\\}\sim\mathfrak{S}\cdot\int_{T\mathcal{B}}\frac{d\mathbf{x}}{\log^{+}(F(\mathbf{x}))},$

where the singular series $\mathfrak{S}$ is defined as a product of local
densities:

$\mathfrak{S}=\prod_{p}\left(\frac{1-p^{-n}\\#\\{\mathbf{x}\in\mathbb{F}_{p}^{n}:F(\mathbf{x})\equiv
0\bmod p\\}}{1-1/p}\right).$

We remark that in the multivariate setting it may be more natural to look at
integers that generate prime ideals $(p)\subset\mathbb{Z}$, rather than
positive primes. For instance,

$F(x,y)=1-x^{2}-y^{2}$

is an absolutely irreducible polynomial with no local restrictions that does
not take any positive prime values. Additionally, we would like to consider
regions more general than boxes. As such, we write down another formulation of
the Bateman-Horn conjecture.

######  Conjecture 1.3.

Let $F(\mathbf{x})$ be a nonconstant irreducible homogeneous polynomial over
$\mathbb{Q}$ in $n$ variables. For any convex compact region
$\mathcal{R}_{0}\subset\mathbb{R}^{n}$, we denote
$T\mathcal{R}_{0}:=\\{t\mathbf{x}:t\in[0,T],\mathbf{x}\in\mathcal{R}_{0}\\}.$
Then, we have that as $T\rightarrow\infty$,

$\\#\\{\mathbf{x}\in T\mathcal{R}_{0}:(F(\mathbf{x}))\subset\mathbb{Z}\textrm{
is a prime
ideal}\\}\sim\mathfrak{S}\cdot\int_{T\mathcal{R}_{0}}\frac{d\mathbf{x}}{\log^{+}(|F(\mathbf{x})|)},$

where

$\mathfrak{S}=\prod_{p}\left(\frac{1-p^{-n}\\#\\{\mathbf{x}\in\mathbb{F}_{p}^{n}:F(\mathbf{x})\equiv
0\bmod p\\}}{1-1/p}\right).$

In this paper, we will prove the multivariate Bateman-Horn conjecture for a
new class of polynomials. Specifically, in Theorem 1.4, we prove a prime
number theorem for the determinant polynomial on $n\times n$ matrices; in
Theorem 1.6, we prove a prime number theorem for the Pffafian on $2n\times 2n$
skew symmetric matrices; in Theorem 1.8, we prove the more involved prime
number theorem for the determinant polynomial on $n\times n$ symmetric
matrices. A crucial ingredient in these proofs is the following uniform upper
bound (Proposition 5.1) for any absolutely irreducible homogeneous polynomial
$F$:

$\limsup_{\mathcal{R}_{0}\subset\mathbb{R}^{n}}\lim_{T\rightarrow\infty}\frac{\\#\\{\mathbf{x}\in
T\mathcal{R}_{0}(\mathbb{Z}):F(\mathbf{x})\textrm{ is
prime}\\}}{|T\mathcal{R}_{0}|/\log(T)}\leq c_{F}.$

Before introducing these results, we will give a brief survey of previous
known results for prime number theorems for multivariate polynomials.

###  1.1. History of multivariate Bateman-Horn conjecture

First, for polynomials $F\in\mathbb{Z}[x_{1},...,x_{n}]$ with $n$ sufficiently
large in the degree $d$, the Hardy-Littlewood circle method can produce such
asymptotics. In particular, for $n>(d-1)2^{d-1}$, the Bateman-Horn conjecture
was proved by Destagnol and Sofos in [10] using the circle method. If $F$ is
diagonal, then an even better bound exists: for $n\geq\lceil
d\log(d)+4.21\rceil$, Brüdern and Wooley’s work [5] on Waring’s problem gives
the Bateman-Horn conjecture.

Beyond the circle method, the other known cases of Bateman-Horn occur when
there is extra algebraic structure. In [15] and [16], Friedlander and Iwaniec
establish that there are infinitely many primes of the form $x^{2}+y^{4}$.
Later, Heath-Brown [18] solved the case of $x^{3}+2y^{3}$; this was then
followed by Heath-Brown and Moroz [19] who finished off the case of
irreducible binary cubic forms. Finally, Maynard [27] used the structure of
norm forms to generalize Heath-Brown and Moroz’s result and handled incomplete
norm forms of degree $n$ in $\geq 15n/22$ variables. We remark that the
results of [15], [16], [18], and [19] all use the structure of norm forms and
the structure of prime factorization in number fields in their respective
proofs. We also note that all of these polynomials factor over
$\overline{\mathbb{Q}}$, whereas our polynomials will be irreducible over
$\overline{\mathbb{Q}}$.

###  1.2. Statements of the main theorems

In this section, we will present prime number theorems for three specific
polynomials.

First, we consider the determinant polynomial $\det$ on $n\times n$ integral
matrices. This polynomial $\det(x_{11},x_{12},...,x_{nn})$ is a non-diagonal
homogeneous polynomial in $n^{2}$ variables of degree $n$ and is irreducible
over $\overline{\mathbb{Q}}.$ For $n\geq 4$, a prime number theorem for this
polynomial is beyond what is known from the circle method. However, by
utilizing results from dynamics, such as the equidistribution of Hecke orbits,
along with an approximation argument, we can achieve the following prime
number theorem.

######  Theorem 1.4.

Define the following prime counting function:

$\pi_{\det}(T):=\\#\\{A\in\textrm{Mat}_{n}(\mathbb{Z}):\|A\|\leq
T,\det(A)\text{ is prime}\\},$

where $\|A\|$ denotes the $L^{\infty}$-norm. As $T\rightarrow\infty$, we have
that

$\pi_{\det}(T)=(1+o(1))\cdot\prod_{j=2}^{n}\zeta(j)^{-1}\int_{\|X\|\leq
T}\frac{1}{\log^{+}(\det(X))}dX,$

where $dX$ is the Euclidean measure on
$\textrm{Mat}_{n}(\mathbb{R})\cong\mathbb{R}^{n^{2}}.$ Moreover, Conjecture
1.3 holds for the determinant polynomial.

######  Remark 1.5.

Our methods lead us to deal with even more general regions, as will be
discussed in §4 and §5. In particular, if
$\mathcal{R}_{0}\subset\textrm{Mat}_{n}(\mathbb{R})$ satisfies that for
$\mathbf{x}\in\mathcal{R}_{0}$, and $t\in[0,1]$, we have that
$t\mathbf{x}\in\mathcal{R}_{0}$, as well as that the boundary of
$\mathcal{R}_{0}$ has Euclidean measure zero, then we have that

$\\#\\{A\in\textrm{Mat}_{n}(\mathbb{Z})\cap T\mathcal{R}_{0}:\det(A)\text{ is
prime}\\}\sim\prod_{j=2}^{n}\zeta(j)^{-1}\int_{T\mathcal{R}_{0}}\frac{1}{\log^{+}(\det(X))}dX.$

This fact will be a consequence of Theorem 3.1 and Theorem 4.1. Some examples
of such $\mathcal{R}_{0}$ include the star-shaped region and compact sets
$\mathcal{R}_{0}$ defining a sector.

We also observe that this result implies that

$\pi_{\det}(T)\sim\frac{1}{2}\cdot\prod_{j=2}^{n}\zeta(j)^{-1}\cdot\frac{2^{n^{2}}T^{n^{2}}}{n\log(T)}.$

Here the factor of $1/2$ comes from the fact that asymptotically half of the
matrices in the box $\|X\|\leq T$ satisfy that $\det(X)>0$.

Second, we consider the Pfaffian polynomial Pff on skew-symmetric $2n\times
2n$ integral matrices. This polynomial is a nondiagonal homogeneous polynomial
in $2n^{2}-n$ variables of degree $n$ and is irreducible over
$\overline{\mathbb{Q}}.$ For $n\geq 5$, a prime number theorem for this
polynomial is beyond what the circle method can currently handle. Again, by
inputting tools from dynamics (in particular Ratner’s theorem), we can
establish a prime number theorem for the Pfaffian.

######  Theorem 1.6.

Let $n\geq 2$. Define the following prime counting function:

$\pi_{\textrm{Pff}}(T):=\\#\\{A\in\textrm{Mat}_{2n}(\mathbb{Z}):\|A\|\leq
T,A^{T}=-A,\textrm{Pff}(A)\text{ is prime}\\},$

where $\|A\|$ denotes the $L^{\infty}$ norm. As $T\rightarrow\infty$, we have
that

$\pi_{\textrm{Pff}}(T)=(1+o(1))\cdot\prod_{\begin{subarray}{c}3\leq n\leq
2n-1\\\ j\text{
odd}\end{subarray}}\zeta(j)^{-1}\cdot\int_{\begin{subarray}{c}\|X\|\leq T\\\
X^{T}=-X\end{subarray}}\frac{1}{\log^{+}(\textrm{Pff}(X))}dX.$

Here $dX$ is the Euclidean measure on
$\textrm{Skew}_{2n}(\mathbb{R})=\\{X\in\textrm{Mat}_{2n}(\mathbb{R}):X^{T}=-X\\}\cong\mathbb{R}^{2n^{2}-n}.$
Moreover, Conjecture 1.3 holds for the Pfaffian.

######  Remark 1.7.

As with the determinant case, we again handle more general regions, as a
consequence of Theorem 3.4 and Theorem 4.1.

Note that both the determinant and the Pfaffian polynomials belong to a more
general dynamical framework; this set up will be discussed further in §2.

Our final example is the determinant polynomial $\det$ on symmetric $n\times
n$ matrices. This polynomial is a nondiagonal homogeneous polynomial in
$\frac{n(n+1)}{2}$ variables of degree $n$ and it is irreducible over
$\overline{\mathbb{Q}}$; furthermore, the variety
$\\{X\in\textrm{Sym}_{n}(\overline{\mathbb{Q}}):\det(X)=0\\}$ is not smooth.
For $n=1$, the Bateman-Horn conjecture is answered by the prime number
theorem. When $n=2$, the determinant becomes a ternary quadratic form and thus
can be handled by the results of Duke in [11]. However, for $n\geq 3$, this
polynomial is currently beyond what is known (from the circle method or other
techniques). The proof of the following theorem makes use of Linnik type
equidistribution as well as the Siegel mass formula.

######  Theorem 1.8.

Let $n\geq 3$. Define the following prime counting function:

$\pi_{\textrm{Sym}}(T):=\\#\\{A\in\textrm{Mat}_{n}(\mathbb{Z}):\|A\|\leq
T,A^{T}=A,\det(A)\text{ is prime}\\},$

where $\|A\|$ denotes the $L^{\infty}$-norm. As $T\rightarrow\infty$, we have
that

$\pi_{\textrm{Sym}}(T)=(1+o(1))\cdot\prod_{\begin{subarray}{c}3\leq j\leq n\\\
j\text{
odd}\end{subarray}}\zeta(j)^{-1}\cdot\int_{\begin{subarray}{c}\|X\|\leq T\\\
X^{T}=X\end{subarray}}\frac{1}{\log^{+}(\det(X))}dX.$

Here $dX$ is the Euclidean measure on
$\textrm{Sym}_{n}(\mathbb{R})=\\{X\in\textrm{Mat}_{n}(\mathbb{R}):X^{T}=X\\}\cong\mathbb{R}^{n(n+1)/2}.$
Moreover, Conjecture 1.3 holds for the determinant polynomial on symmetric
matrices.

######  Remark 1.9.

Again, we can handle more general regions than boxes; see Theorem 4.1 and §7.

Moreover, we prove a more precise result depending on the signature of the
symmetric matrices involved; this is discussed further in §7.

######  Remark 1.10.

The question of prime determinants for the determinant polynomial on
$\textrm{Mat}_{n}(\mathbb{Z})$ and $\textrm{Sym}_{n}(\mathbb{Z})$ was raised
by Kovaleva in her insightful investigation in [22]. Kovaleva counts in [22]
matrices with squarefree, instead of prime, determinants and resolves the
variant question of counting $n\times n$ symmetric integer matrices with a
squarefree minor of size $n-r$ for $r>0$.

###  1.3. Outline of the paper

First, in §2 we describe the general framework that all of our examples belong
to – they are all almost Hardy-Littlewood systems (Definition 2.4). We present
a general prime number theorem (1) under certain conditions and relate the
problem of counting primes to the Linnik problem. Additionally, in §2 we will
discuss the Tamagawa measure, the “Bateman-Horn measure”, and how these two
measures relate to the Euclidean measure.

In §3, we prove a prime number theorem for the determinant polynomial on
integral matrices and a prime number theorem for the Pfaffian on integral
skew-symmetric matrices for particularly structured regions; namely, we look
at cones.

In §4, we start our procedure for turning our prime number theorem on cones
into a theorem on boxes (and more general regions). We introduce the notion of
an $\epsilon$-cutting, which is an approximation of the desired region by
cones. The idea of $\epsilon$-cuttings comes from the principle of Riemannian
integration – one can approximate a general compact region $\mathcal{R}$ using
sequences of cones of smaller and smaller width.

In §5, we introduce the second ingredient of our procedure: an upper bound
sieve on the part of the box that is not covered by the approximation by
cones. Section 5 does not make use of the dynamics framework and it applies to
a general homogeneous irreducible polynomial. It yields an upper bound for
Conjecture 1.3, which is off by a constant independent of $\mathcal{R}_{0}.$
Finally in §5, we present the proofs of Theorem 1.4 and Theorem 1.6.

In §6, we establish the necessary background on the Siegel mass formula to
prove Theorem 1.8, which is subsequently proved in §7. Finally, in §8, we
comment on variant questions that our methods can handle.

##  2\. General framework of invariant polynomials for homogeneous spaces

In this section, we aim to put our choices of polynomials in a general
framework – they are the invariant polynomials of homogeneous spaces.

###  2.1. The Linnik Problem

The proofs of Theorems 1.4, 1.6, and 1.8 rely on the resolution of the Linnik
problem in the respective set ups. In this subsection, we explain the Linnik
problem.

Let $F(\mathbf{x})\in\mathbb{Z}[\mathbf{x}]$ be a general polynomial in $n$
variables, let

$V_{m}=\\{\mathbf{x}:F(\mathbf{x})=m\\}$

be the level sets of $F$, and let $V_{m}(\mathbb{Z})$ denote the integer
points of $V_{m}$. The Linnik problem studies the equidistribution of
$V_{m}(\mathbb{Z})$ when projected onto $V_{1}(\mathbb{R})$ as
$m\rightarrow\infty$. We list some examples.

Linnik [24] used his ergodic method to prove (conditional on $m$ being a
quadratic residue for any fixed odd prime) that for
$F(\mathbf{x})=x_{1}^{2}+x_{2}^{2}+x_{3}^{2},$ for $\mu$ the normalization of
the Euclidean measure on $\mathbf{S}^{2}$, and for
$\Omega\subset\mathbf{S}^{2}$ a subset with boundary of measure zero, the
following holds:

$\lim_{m\rightarrow\infty}\frac{\\#V_{m}(\mathbb{Z})\cap
m^{\frac{1}{2}}\Omega}{\\#V_{m}(\mathbb{Z})}=\mu(\Omega).$

This result was later proved unconditionally by Duke in [11] using bounds on
the Fourier coefficients of half-weight modular forms proved by Iwaniec in
[20].

In [25], Linnik and Skubenko solve the corresponding problem for the
determinant polynomial on $\textrm{GL}_{n}(\mathbb{R})$. Notice that unlike
the case of the sphere, $V_{m}(\mathbb{Z})$ is infinite for every
$m\in\mathbb{N}$, which can be understood as a consequence of
$\textrm{SL}_{n}(\mathbb{R})$ having infinite measure and hence it is
necessary to intersect with some compact set. We refer the reader to Sarnak’s
1990 ICM talk [30] for the connection between this problem and the
equidistribution of Hecke orbits. Sarnak shows the following result, which is
effective with a power saving: let $\Omega\subset\textrm{SL}_{n}(\mathbb{R})$
be a “nice” compact set (as defined in Definition 2.2), $F$ be the determinant
polynomial, and $\mu$ the Euclidean measure on $\textrm{SL}_{n}(\mathbb{R})$
(induced from $\textrm{GL}_{n}(\mathbb{R})$). Then, as $m\rightarrow\infty:$

$\\#V_{m}(\mathbb{Z})\cap
m^{\frac{1}{n}}\Omega\sim\prod_{j=2}^{n}\zeta(j)^{-1}\cdot
G_{n}(m)\mu(\Omega),$

where for $m=\prod_{i=1}^{r}p_{i}^{e_{i}},$ we define

$G_{n}(m)=\prod_{i=1}^{r}\frac{(p_{i}^{e_{i}+1}-1)(p_{i}^{e_{i}+2}-1)\dots(p_{i}^{e_{i}+n-1}-1)}{(p_{i}-1)(p_{i}^{2}-1)\dots(p_{i}^{n-1}-1)}.$

A robust approach to these Linnik type problems is to use Ratner’s work on
measure rigidity [29]. We will make use of the work of Eskin and Oh [14].

###  2.2. The general framework

Let $G$ denote some semisimple real algebraic group defined over $\mathbb{Q}$
and let $\rho:G\rightarrow\textrm{GL}(V)$ be a rational representation of $G$
on a finite vector space $V$. Assume that there is some polynomial $F$ that is
invariant under the right action of $\varrho$, i.e.

$F(v\varrho(g))=F(v),\forall g\in G,$

and that $F$ is irreducible over $\overline{\mathbb{Q}}$. We denote the level
sets of $F$ by

$V_{m}:=\\{v\in V:F(v)=m\\}.$

For some $v_{0}\in V$, let $H=\text{stab}_{v_{0}}$ denote the stabilizer of
$v_{0}$ inside $G$. We require that the set $(V,G,H,F)$ satisfies the
following strong assumptions:

  * •

$V_{1}=v_{0}G$ for some vector $v_{0}\in V(\mathbb{R})$, i.e. $V_{1}$ can be
identified with the homogeneous space

$V_{1}\cong G/H,$

  * •

for each $m$, there is a $\lambda_{m}\in\mathbb{R}$ such that
$V_{m}=\lambda_{m}V_{1}$,

  * •

$V_{1}(\mathbb{Q})\neq\emptyset$, i.e. $V_{1}$ contains a rational point,

  * •

$H(\mathbb{R})\subset G(\mathbb{R})$ is a connected, simply connected, and
semisimple subgroup. We also assume that it is a proper maximal subgroup of
$G(\mathbb{R})$ and contains no compact factors,

  * •

if $m_{0}\in\mathbb{Z}$ satisfies that $\\#\\{v\in
V(\mathbb{Z}):F(v)=m_{0}\\}\neq 0$, then we must have

$\\#\\{m\in\mathbb{Z}:\lambda_{m}^{-1}V_{m}(\mathbb{Z})=\lambda_{m_{0}}^{-1}V_{m_{0}}(\mathbb{Z})\\}<\infty.$

When $V_{1}\cong G/H$ is an affine symmetric space, these varieties satisfy
the Hasse principle, as was shown by Borovoi in [4]; in more generality we are
assuming the existence of an initial rational point. The fact that the
stabilizer $H(\mathbb{R})$ is connected and semisimple ensures the existence
and uniqueness of the Tamagawa measure on the level sets of $F$. The final two
conditions are what is necessary in order to apply Theorem 1.9 of Oh [28].

######  Remark 2.1.

We note that the case $V_{1}=\bigcup_{i}v_{i}G$ is a finite union of orbits,
where the stabilizers of $v_{i}$ satisfy all of the above conditions, is
essentially the same and we discuss the single orbit case here only to make
the exposition simpler.

######  Definition 2.2.

We will call a compact set $\Omega\subset V_{1}(\mathbb{R})$ nice if its
boundary has measure zero on $V_{1}(\mathbb{R}).$ For a nice compact subset
$\Omega\subset V_{1}(\mathbb{R})$, we define the cone of $\Omega$ of height
$T$ as the following region:

$T\Omega:=\\{v=t\omega:t\in[0,T],\omega\in\Omega\\}.$

Under the assumptions listed above, Eskin and Oh prove in [14] that for any
$\Omega$ nice compact subset of $V_{1}(\mathbb{R})$ we have

$\\#V_{m}(\mathbb{Z})\cap
m^{\frac{1}{n}}\Omega\sim\omega_{m}\mu(\Omega),\,m\rightarrow\infty,$

where $\mu$ is some $G-$invariant Borel measure on $V_{1}$ and $\omega_{m}$ is
defined as follows: for any arithmetic lattice $\Gamma$ in $G(\mathbb{Q})$
with $V_{m}(\mathbb{Z})\Gamma\subset V_{m}(\mathbb{Z})$ for all
$m\in\mathbb{Z}$, we have

$\omega_{m}(\Gamma)=\frac{\sum\limits_{\xi\Gamma\subset
V_{m}(\mathbb{Z})}\mu_{H}((H\cap g_{\xi}^{-1}\Gamma g_{\xi})\backslash
H)}{\mu_{G}(\Gamma\backslash G)},$

where

  * •

the sum is over all disjoint orbits $\xi\Gamma$ inside $V_{m}(\mathbb{Z})$;
this sum is finite as a consequence of the theorem of Borel and Harish-Chandra
[2],

  * •

$g_{\xi}$ is a point of $G$ such that $v_{0}=(m^{-\frac{1}{n}}\xi)g_{\xi}$,

  * •

the Borel measure $\mu$ and the Haar measures $\mu_{G},\mu_{H}$ are compatible
in the sense of Weil [33].

In [28], Oh proves that $\omega_{m}$ does not depend on the choice of $\Gamma$
and thus the notation is consistent. An immediate corollary is that for nice
compact sets $\Omega_{1},\Omega_{2}\subset V_{1}$:

$\frac{\\#V_{m}(\mathbb{Z})\cap
m^{\frac{1}{n}}\Omega_{1}}{\\#V_{m}(\mathbb{Z})\cap
m^{\frac{1}{n}}\Omega_{2}}\sim\frac{\mu(\Omega_{1})}{\mu({\Omega_{2}})}.$

Finally, Oh computes the constant $\omega_{m}$ in terms of local masses and
derives the following, which will be crucial for our analysis:

######  Theorem 2.3 (Oh, [28]).

Let $V$ be a linear space with a given $\mathbb{Z}$-structure satisfying the
above conditions. Then for any nice compact set $\Omega\subset
V_{1}(\mathbb{R})$, as $m\rightarrow\infty$,

$\\#V_{m}(\mathbb{Z})\cap\mathcal{R}(m^{1/\deg(F)},\Omega)=\mu_{\infty}(\Omega)\cdot
m^{\dim(V)/\deg(F)-1}\cdot\mathfrak{S}_{m}\cdot(1+o_{\Omega}(1)),$

where the singular series is given by

$\mathfrak{S}_{m}=\prod_{q}\lim_{k\rightarrow\infty}\frac{V_{m}(\mathbb{Z}/q^{k}\mathbb{Z})}{q^{k\dim(V_{m})}},$

and the dependence on the compact set is given by

$\mu_{\infty}(\Omega)=\lim_{\epsilon\rightarrow
0}\frac{\textrm{vol}(\\{v\in[1-\epsilon,1+\epsilon]\Omega\\})}{2\epsilon},$

as computed by Borovoi and Rudnick in [3].

This result of Oh implies that under the conditions above on $(V,G,H,F)$, we
can conclude that $(V,F)$ satisfies the predictions made by Hardy and
Littlewood for rational points on varieties; in particular, we will call these
systems Hardy-Littlewood systems.

######  Definition 2.4.

A linear space $V$ with polynomial $F$ is a Hardy-Littlewood system if it
satisfies that as $m\rightarrow\infty$,

$N(m,\Omega)=\mu_{\infty}(\Omega)\cdot
m^{\dim(V)/\deg(F)-1}\cdot\mathfrak{S}_{m}\cdot(1+o_{\Omega}(1)),$

where the singular series is given by

$\mathfrak{S}_{m}=\prod_{q}\lim_{k\rightarrow\infty}\frac{\\#V_{m}(\mathbb{Z}/q^{k}\mathbb{Z})}{q^{k\dim(V_{m})}},$

and the dependence on the compact set is given by

$\mu_{\infty}(\Omega)=\lim_{\epsilon\rightarrow
0}\frac{\textrm{vol}(\\{v\in[1-\epsilon,1+\epsilon]\Omega\\})}{2\epsilon}.$

Here the volume denoted is the Euclidean volume.

The above asymptotic result on the count of integral points on $V_{m}$ will
allow us to derive a prime number theorem on cones by summing over the level
sets with prime values. In particular, if $(V,F)$ is a Hardy-Littlewood system
and the singular series $\mathfrak{S}_{m}$ satisfies that for all sufficiently
large primes $p$,

$\mathfrak{S}_{p}=\mathfrak{S}_{V,F}(1+O(p^{-\delta})),$

where $\mathfrak{S}_{V,F}$ is the singular series from Conjecture 1.2 and
$\delta>0$ is a positive constant, then the following prime number theorem
holds for any nice compact set $\Omega\subset V_{1}(\mathbb{R})$:

(1) $\displaystyle\\#\\{v\in V(\mathbb{Z})\cap T\Omega:F(v)\text{ is
prime}\\}\sim$
$\displaystyle\mathfrak{S}_{V,F}\cdot\int_{T\Omega}\frac{1}{\log^{+}(F(v))}dv.$

The proof of this fact follows exactly the proof presented in §3.2 for the
determinant polynomial and the Pfaffian.

###  2.3. Relating the measures

So far, we have discussed the measure $\mu_{\infty}$ on $V_{1}$, which appears
in Theorem 2.3, and the Euclidean measure, which appears as the archimedean
part of the Bateman-Horn conjecture. In this subsection, we wish to provide a
relation between these two measures. In order to establish this connection, we
utilize the co-area formula.

######  Lemma 2.5.

For $\Omega$ a nice compact connected subset of $V_{1}(\mathbb{R})$, we have
that as $T\rightarrow\infty$,

$\int_{T\Omega}\frac{1}{\log^{+}(F(x))}dx=\mu_{\infty}(\Omega)\cdot\frac{T^{\dim(V)}}{\dim(V)\log(T)}(1+o(1)).$

###### Proof.

From the computations of Borovoi and Rudnick in [3], we know that

$\mu_{\infty}(\Omega)=\lim_{\epsilon\rightarrow
0}\frac{1}{2\epsilon}\int_{[1-\epsilon,1+\epsilon]\Omega}dx,$

where $dx$ is the Euclidean measure on $V(\mathbb{R}).$ The co-area formula
then allows us to rewrite this formula in terms of an integral over $\Omega$:

$\displaystyle\int_{[1-\epsilon,1+\epsilon]\Omega}dx$
$\displaystyle=\int_{1-\epsilon}^{1+\epsilon}\int_{r^{1/\deg(F)}\Omega}|\nabla
F(y)|^{-1}dydr$
$\displaystyle=\int_{1-\epsilon}^{1+\epsilon}r^{\frac{\dim(V)}{\deg(F)}-1}\int_{\Omega}|\nabla
F(y)|^{-1}dydr$ $\displaystyle=\left(\int_{\Omega}|\nabla
F(y)|^{-1}dy\right)\cdot((1+\epsilon)^{\dim(V)/\deg(F)}-(1-\epsilon)^{\dim(V)/\deg(F)})\cdot\frac{\deg(F)}{\dim(V)}.$

Hence, taking the limit as $\epsilon\rightarrow 0$, we have that

$\mu_{\infty}(\Omega)=\int_{\Omega}|\nabla F(y)|^{-1}dy.$

On the other hand, we consider the archimedean part of the Bateman-Horn
conjecture. Applying the co-area formula again, we have that as
$T\rightarrow\infty$:

$\displaystyle\int_{T\Omega}\frac{1}{\log^{+}(F(x))}dx$
$\displaystyle=\int_{2}^{T^{\deg(F)}}\frac{1}{\log(r)}\int_{r^{1/\deg(F)}\Omega}|\nabla
F(y)|^{-1}dydr$
$\displaystyle=\int_{2}^{T^{\deg(F)}}\frac{r^{\frac{\dim(V)}{\deg(F)}-1}}{\log(r)}\int_{\Omega}|\nabla
F(y)|^{-1}dydr$ $\displaystyle=\left(\int_{\Omega}|\nabla
F(y)|^{-1}dy\right)\cdot\frac{T^{\dim(V)}}{\dim(V)\log(T)}(1+o(1)).$

∎

######  Remark 2.6.

We recall that in Oh [28], the measure $\mu_{\infty}$ is a gauge form, a
nowhere zero regular differential form of maximal degree, which is used to
construct the unique Tamagawa measure on $V_{1}(\mathbb{A})$. Hence, the co-
area formula gives us a precise expression for this Gauge form $d\mu_{\infty}$
on $V_{1}:$

$d\mu_{\infty}=|\nabla F(y)|^{-1}dy.$

From now on, we will be calling $\mu_{\infty}$ the Bateman Horn measure.

##  3\. A prime number theorem on cones

In this section, we prove a prime number theorem on particularly structured
regions for the determinant polynomial on $\textrm{Mat}_{n}(\mathbb{R})$; in
particular we look at cones. The reason for that is that we can solve the
Linnik problem for this space (as mentioned in §2.1). We will explain in the
end of the section that the case of the Pfaffian follows the same lines.

For a nice compact subset $\Omega\subset
V_{1}(\mathbb{R})=\textrm{SL}_{n}(\mathbb{R})$, we define

$\pi_{\textrm{Mat}_{n},\det}(T,\Omega)=\\#\\{A\in
T\Omega\cap\textrm{Mat}_{n}(\mathbb{Z}):\text{ $\det A$ is prime}\\}.$

######  Theorem 3.1.

Let $\Omega$ be a nice compact subset of $\textrm{SL}_{n}(\mathbb{R}).$ Then
as $T\rightarrow\infty$, the Bateman-Horn conjecture holds on $T\Omega$:

$\pi_{\textrm{Mat}_{n},\det}(T,\Omega)\sim\mathfrak{S}_{\textrm{Mat}_{n},\det}\cdot\int_{T\Omega}\frac{1}{\log^{+}(\det(\mathbf{x}))}d\mathbf{x},$

where

$\mathfrak{S}_{\textrm{Mat}_{n},\det}=\prod_{p}\left(\frac{1-p^{-n^{2}}\\#\\{\mathbf{x}\in\textrm{Mat}_{n}(\mathbb{F}_{p}):\det(\mathbf{x})\equiv
0\bmod p\\}}{(1-1/p)}\right).$

Next, we recall the following corollary of Theorem 2.3, as proved by Oh in
Example 4.1 of [28]. In particular, the following shows that
$(\textrm{Mat}_{n},\det)$ is a Hardy-Littlewood system and gives us an
asymptotic count for the integral points on the level sets.

######  Theorem 3.2 (Oh [28], Example 4.1).

For any nice compact subset $\Omega\subset
V_{1}(\mathbb{R})=\textrm{SL}_{n}(\mathbb{R})$, as $m\rightarrow\infty$:

$\\#V_{m}(\mathbb{Z})\cap
m^{1/n}\Omega=(1+o_{\Omega}(1))\cdot\mu_{\infty}(\Omega)\cdot
m^{n-1}\cdot\mathfrak{S}_{m},$

where

$\mathfrak{S}_{m}=\prod_{q\text{
prime}}\lim_{k\rightarrow\infty}\frac{\\#\\{X\in\textrm{Mat}_{n}(\mathbb{Z}/q^{k}\mathbb{Z}):\det(X)=m\\}}{q^{k(n^{2}-1)}}$

and $\mu_{\infty}$ is the Bateman-Horn measure.

We note that the singular series in Theorem 3.1 differs in definition from the
singular series in Theorem 3.2. In the next section, we show that these
products are not so different.

###  3.1. Stability of the singular series

We remark that our desired singular series from the Bateman-Horn conjecture
(Conjecture 1.3) satisfies:

$\mathfrak{S}_{\textrm{Mat}_{n},\det}=\prod_{p}\frac{p^{-n^{2}+1}\\#\\{v\in\textrm{Mat}_{n}(\mathbb{F}_{p}):\det(v)\in\mathbb{F}_{p}^{\times}\\}}{\varphi(p)}.$

On the other hand, we also have the following Hardy-Littlewood singular series
for any $m\in\mathbb{Z}$:

$\mathfrak{S}_{m}=\prod_{q\text{
prime}}\lim_{k\rightarrow\infty}\frac{\\#\\{v\in\textrm{Mat}_{n}(\mathbb{Z}/q^{k}\mathbb{Z}):\det(v)=m\\}}{q^{k(n^{2}-1)}}.$

We will now show that these two quantities are similar when we specify $m$ to
be a prime $p$.

######  Lemma 3.3.

For any sufficiently large prime $p$, the singular series $\mathfrak{S}_{p}$
coming from the Hardy-Littlewood property of $(\textrm{Mat}_{n},\det)$
satisfies

$\mathfrak{S}_{p}=\mathfrak{S}_{\textrm{Mat}_{n},\det}\cdot(1+O(p^{-\frac{1}{2}})).$

###### Proof.

We first notice the following local equidistribution property of
$\textrm{Mat}_{n}$: For any prime power $q^{k}$ and any $0\leq a,b<q^{k}$ with
$(a,q^{k})=(b,q^{k})$, we have that

(2)
$\displaystyle\\#\\{v\in\textrm{Mat}_{n}(\mathbb{Z}/q^{k}\mathbb{Z}):\det(v)=a\\}$
$\displaystyle=\\#\\{v\in\textrm{Mat}_{n}(\mathbb{Z}/q^{k}\mathbb{Z}):\det(v)=b\\}.$

It is then easy to see that for $q$ a prime that is not $p$:

$\\#\\{v\in\textrm{Mat}_{n}(\mathbb{F}_{q}):\det(v)=p\\}=\frac{\\#\\{v\in\textrm{Mat}_{n}(\mathbb{F}_{q}):\det(v)\in\mathbb{F}_{q}^{\times}\\}}{\varphi(q)},$

and

$\lim_{k\rightarrow\infty}\frac{\\#\\{v\in\textrm{Mat}_{n}(\mathbb{Z}/q^{k}\mathbb{Z}):\det(v)=p\\}}{q^{k(n^{2}-1)}}=\frac{\\#\\{v\in\textrm{Mat}_{n}(\mathbb{F}_{q}):\det(v)\in\mathbb{F}_{q}^{\times}\\}}{\varphi(q)}.$

We recognize the right hand side as the $q$-factor in the Euler product of
$\mathfrak{S}_{\textrm{Mat}_{n},\det}.$ We now prove Eq. 2: We do this for
$a=k=1,$ since the general case is similar. We consider the map

$\\{A\in\textrm{Mat}_{n}(\mathbb{Z}/q\mathbb{Z}):\det(A)=1\\}\rightarrow\\{B\in\textrm{Mat}_{n}(\mathbb{Z}/q\mathbb{Z}):\det(B)=b\\}$

given by

$A\mapsto\begin{pmatrix}b&0&...&0\\\ 0&1&...&0\\\
\vdots&\vdots&\vdots&\vdots\\\ 0&0&...&1\end{pmatrix}A$

for $A$ with $\det(A)=1$. Since $\gcd(b,q)=1$, this matrix
$\text{diag}(b,1,...,1)$ is invertible in $\textrm{Mat}_{n}(\mathbb{F}_{q})$,
so this gives us a bijection between the two sets. Thus, we get the
equidistribution property. It follows that the local density at $q$ of the
singular series $\mathfrak{S}_{p}$ and $\mathfrak{S}_{\textrm{Mat}_{n},\det}$
match at all prime $q\neq p$.

So, it remains to check that the factors almost match at the prime $p$. Since
$\\{\det(v)=0\\}$ is an absolutely irreducible variety over
$\overline{\mathbb{Q}}$, we know that for $p$ sufficiently large this will be
an absolutely irreducible variety over $\mathbb{F}_{p}$. Hence, the Lang-Weil
bound [23] implies that the following holds:

(3)
$\\#\\{v\in\textrm{Mat}_{n}(\mathbb{Z}/p\mathbb{Z}):\det(v)=p\\}=p^{\dim(V)-1}(1+O(p^{-1/2})).$

Thus, we know that

$\frac{p^{-\dim(V)+1}\\#\\{v\in
V(\mathbb{F}_{p}):F(v)\in\mathbb{F}_{p}^{\times}\\}}{\varphi(p)}=1+O(p^{-\frac{3}{2}}).$

On the other hand, from the equidistribution property (2), for $k\geq 2$ we
have that

$\lim_{k\rightarrow\infty}\frac{\\#\\{v\in\textrm{Mat}_{n}(\mathbb{Z}/p^{k}\mathbb{Z}):\det(v)\equiv
p\bmod
p^{k}\\}}{p^{k(n^{2}-1)}}=\frac{\\#\\{v\in\textrm{Mat}_{n}(\mathbb{Z}/p^{2}\mathbb{Z}):\det(v)\equiv
p\bmod p^{2}\\}}{p^{2(n^{2}-1)}}.$

Evaluating the right-hand side, we have that the expression equals:

$\displaystyle\frac{\\#\textrm{Mat}_{n}(\mathbb{Z}/p^{2}\mathbb{Z})}{\varphi(p)p^{2(n^{2}-1)}}-\frac{\\#\\{v\in\textrm{Mat}_{n}(\mathbb{Z}/p^{2}\mathbb{Z}):\det(v)\in(\mathbb{Z}/p^{2}\mathbb{Z})^{\times}\\}}{\varphi(p)p^{2(n^{2}-1)}}-\frac{\\#\\{v\in\textrm{Mat}_{n}(\mathbb{Z}/p^{2}\mathbb{Z}):\det(v)\equiv
0\bmod p^{2}\\}}{\varphi(p)p^{2(n^{2}-1)}}$
$\displaystyle=p+1+O(p^{-1})-\frac{p\\#\\{v\in\textrm{Mat}_{n}(\mathbb{F}_{p}):\det(v)\in\mathbb{F}_{p}^{\times}\\}}{\varphi(p)p^{n^{2}-1}}-\frac{\\#\\{v\in\textrm{Mat}_{n}(\mathbb{Z}/p^{2}\mathbb{Z}):\det(v)\equiv
0\bmod p^{2}\\}}{\varphi(p)p^{2(n^{2}-1)}}$
$\displaystyle=1-\frac{\\#\\{v\in\textrm{Mat}_{n}(\mathbb{Z}/p^{2}\mathbb{Z}):\det(v)\equiv
0\bmod p^{2}\\}}{p^{2n^{2}-1}}+O(p^{-\frac{1}{2}}).$

In the second equality, we have used (2) to handle the middle term. In the
third equality, we have applied (3). It now suffices to show that for $p$
sufficiently large,

(4) $\\#\\{v\in\textrm{Mat}_{n}(\mathbb{Z}/p^{2}\mathbb{Z}):\det(v)\equiv
0\bmod p^{2}\\}=O(p^{2n^{2}-2}),$

since it will follow that

$\lim_{k\rightarrow\infty}\frac{\\#\\{v\in\textrm{Mat}_{n}(\mathbb{Z}/p^{k}\mathbb{Z}):\det(v)\equiv
p\bmod p^{k}\\}}{p^{k(n^{2}-1)}}=1+O(p^{-1/2}).$

Let $H=\\{v\in\textrm{Mat}_{n}(\mathbb{F}_{p}):\det(v)\equiv 0\bmod p\text{
and $\nabla{\det}(v)\neq\b{0}\bmod p$}\\}$ denote the points of
$\\{v\in\textrm{Mat}_{n}(\mathbb{F}_{p}):\det(v)\equiv 0\bmod p\\}$ that admit
a Hensel lifting (see [7]), and let
$H^{c}=\\{v\in\textrm{Mat}_{n}(\mathbb{F}_{p}):\det(v)\equiv 0\bmod p\text{
and $\nabla{\det}(v)=\b{0}\bmod p$}\\}$ denote its complement. We then have

$\\#\\{v\in\textrm{Mat}_{n}(\mathbb{Z}/p^{2}\mathbb{Z}):\det(v)\equiv 0\bmod
p^{2}\\}\leq\\#H\cdot p^{n^{2}-1}+\\#H^{c}\cdot p^{n^{2}}.$

By the trivial bound we obtain

$\\#H\leq\\#\\{v\in\textrm{Mat}_{n}(\mathbb{F}_{p}):\det(v)\equiv 0\bmod
p\\}\ll p^{n^{2}-1}.$

Therefore, in order to show (4), it suffices to show

$\\#H^{c}=O(p^{n^{2}-2}).$

Indeed, since $\\{v\in\textrm{Mat}_{n}(\mathbb{C}):\det(v)=0\\}$ is
geometrically irreducible over $\mathbb{C}$, we know that
$\\{v\in\textrm{Mat}_{n}(\mathbb{F}_{p}):\det(v)\equiv 0\bmod p\\}$ is
geometrically irreducible over $\overline{\mathbb{F}_{p}}$ for all
sufficiently large primes $p$. So, the codimension of the singular locus of
$\\{v\in\textrm{Mat}_{n}(\mathbb{F}_{p}):\det(v)=0\bmod p\\}$ will be at least
one for such primes. As such, we know that
$\dim(\\{v\in\textrm{Mat}_{n}(\mathbb{F}_{p}):\det(v)\equiv 0\bmod
p,\nabla\det(v)=\b{0}\bmod p\\})\leq n^{2}-2.$ Hence, we have that

$\\#H^{c}=\left|\\{v\in\textrm{Mat}_{n}(\mathbb{F}_{p}):\det(v)\equiv 0\bmod
p,\textrm{ and }\nabla\det(v)=\b{0}\bmod p\\}\right|\ll p^{n^{2}-2}.$

This completes the proof of (4).

Finally, we put all the local densities together. For $p\neq q$, we have that

$\lim_{k\rightarrow\infty}\frac{\\#\\{v\in\textrm{Mat}_{n}(\mathbb{Z}/q^{k}\mathbb{Z}):\det(v)=p\\}}{q^{k(n^{2}-1)}}=\frac{\\#\\{v\in\textrm{Mat}_{n}(\mathbb{F}_{q}):\det(v)\in\mathbb{F}_{q}^{\times}\\}}{\varphi(q)}.$

On the other hand, the local density at $p$ is given by

$\lim_{k\rightarrow\infty}\frac{\\#\\{v\in\textrm{Mat}_{n}(\mathbb{Z}/p^{k}\mathbb{Z}):\det(v)\equiv
p\bmod p^{k}\\}}{p^{k(n^{2}-1)}}=1+O(p^{-1/2}).$

Thus, we have that

$\mathfrak{S}_{p}=\mathfrak{S}_{\textrm{Mat}_{n},\det}(1+O(p^{-1/2})).$

∎

###  3.2. Counting primes in cones

Using the results of this section we are now in a position to prove Theorem
3.1:

###### Proof of Theorem 3.1.

From Theorem 3.2, for $T$ sufficiently large, we know that

$\displaystyle\pi_{\textrm{Mat}_{n},\det}(T\Omega)$
$\displaystyle=(1+o_{\Omega}(1))\mu_{\infty}(\Omega)\sum_{p\leq
T^{n}}p^{n-1}\cdot\mathfrak{S}_{p}$
$\displaystyle=(1+o_{\Omega}(1))\mu_{\infty}(\Omega)\mathfrak{S}_{\textrm{Mat}_{n},\det}\sum_{p\leq
T^{n}}p^{n-1}(1+O(p^{-1/2}))$
$\displaystyle=(1+o_{\Omega}(1))\mu_{\infty}(\Omega)\mathfrak{S}_{\textrm{Mat}_{n},\det}\cdot\frac{T^{n}}{n^{2}\log(T)}$
$\displaystyle=(1+o_{\Omega}(1))\mathfrak{S}_{\textrm{Mat}_{n},\det}\int_{T\Omega}\frac{dx}{\log^{+}(\det(x))}.$

The second equality follows from Lemma 3.3 and the fourth equality comes from
Lemma 2.5. We note that $\mathfrak{S}_{\textrm{Mat}_{n},\det}$ is the singular
series given by the Bateman-Horn conjecture. ∎

Finally, we remark that since

$\\#\\{v\in\textrm{Mat}_{n}(\mathbb{F}_{p}):\det(v)\in\mathbb{F}_{p}^{\times}\\}=\prod_{j=0}^{n-1}(p^{n}-p^{j}),$

we can evaluate the singular series as

$\mathfrak{S}_{\textrm{Mat}_{n},\det}=\prod_{j=2}^{n}\zeta(j)^{-1}.$

###  3.3. Modifications for the Pfaffian polynomial

The analysis for the Pfaffian on $2n\times 2n$ skew-symmetric matrices follows
along the same lines as the proof of Theorem 3.1 described above. In this
subsection, we prove the corresponding prime number theorem on cones.

Let $V=\textrm{Skew}_{2n}=\\{X\in\textrm{Mat}_{2n}:X^{T}=-X\\}$. For a nice
compact subset $\Omega\subset V_{1}(\mathbb{R})$, we define

$\pi_{\textrm{Skew}_{2n},\textrm{Pff}}(T,\Omega)=\\#\\{A\in
T\Omega\cap\textrm{Skew}_{2n}(\mathbb{Z}):\textrm{Pff}(A)\text{ is prime}\\}.$

######  Theorem 3.4.

Let $\Omega$ be a nice compact subset of
$V_{1}(\mathbb{R})=\\{X\in\textrm{SL}_{2n}(\mathbb{R}):X^{T}=-X\\}.$ Then as
$T\rightarrow\infty$, the Bateman-Horn conjecture holds on $T\Omega$:

$\pi_{\textrm{Skew}_{2n},\textrm{Pff}}(T,\Omega)\sim\mathfrak{S}_{\textrm{Skew}_{2n},\textrm{Pff}}\cdot\int_{T\Omega}\frac{1}{\log^{+}(\textrm{Pff}(\mathbf{x}))}d\mathbf{x},$

where

$\mathfrak{S}_{\textrm{Skew}_{2n},\textrm{Pff}}=\prod_{p}\left(\frac{1-p^{-(2n^{2}-n)}\\#\\{\mathbf{x}\in\textrm{Skew}_{2n}(\mathbb{F}_{p}):\textrm{Pff}(x)\equiv
0\bmod p\\}}{(1-1/p)}\right).$

We start by noting that Theorem 2.3 also implies that this system is Hardy-
Littlewood. In particular, we have that by Example 4.2 of Oh [28]:

######  Theorem 3.5 (Oh [28], Example 4.2).

For any nice compact subset $\Omega\subset V_{1}(\mathbb{R})$, as
$m\rightarrow\infty$:

$\\#V_{m}(\mathbb{Z})\cap
m^{1/n}\Omega=(1+o_{\Omega}(1))\cdot\mu_{\infty}(\Omega)\cdot
m^{2n-2}\cdot\mathfrak{S}_{m},$

where

$\mathfrak{S}_{m}=\prod_{q\text{
prime}}\lim_{k\rightarrow\infty}\frac{\\#\\{X\in\textrm{Skew}_{2n}(\mathbb{Z}/q^{k}\mathbb{Z}):\textrm{Pff}(X)=m\\}}{q^{k(n^{2}-1)}}$

and $\mu_{\infty}$ is the Bateman-Horn measure.

Next, we want to prove that these singular series
$\mathfrak{S}_{\textrm{Skew}_{2n},\textrm{Pff}}$ and $\mathfrak{S}_{m}$ are
similar when $m=p$ is a large prime.

######  Lemma 3.6.

For any sufficiently large prime $p$, we have that

$\mathfrak{S}_{p}=\mathfrak{S}_{\textrm{Skew}_{2n},\textrm{Pff}}\cdot(1+O(p^{-1/2})).$

###### Proof.

We note that, in the proof of Lemma 3.3, the only aspect that was specific to
$(\textrm{Mat}_{n},\det)$ was the local equidistribution property (2). So, it
suffices to prove a local equidistribution property for
$(\textrm{Skew}_{2n},\textrm{Pff}):$ For any $0\leq a,b<q^{k}$ and
$\gcd(a,q^{k})=\gcd(b,q^{k})$, we have that

$\\#\\{X\in\textrm{Skew}_{2n}(\mathbb{Z}/q^{k}\mathbb{Z}):\textrm{Pff}(X)=a\\}=\\#\\{X\in\textrm{Skew}_{2n}(\mathbb{Z}/q^{k}\mathbb{Z}):\textrm{Pff}(X)=b\\}.$

Again, we show this fact when $q$ is a prime and $(b,q)=1$:

(5) $\\#\\{X\in\textrm{Skew}_{2n}(\mathbb{F}_{q}):\textrm{Pff}(X)\equiv 1\bmod
q\\}=\\#\\{X\in\textrm{Skew}_{2n}(\mathbb{F}_{q}):\textrm{Pff}(X)\equiv b\bmod
q\\}.$

The cases when $a\neq 1$ and $k\neq 1$ can be handled by the same proof.

We consider the map

$\varphi:\\{A\in\textrm{Skew}_{2n}(\mathbb{F}_{q}):\textrm{Pff}(A)\equiv
1\bmod
q\\}\rightarrow\\{B\in\textrm{Skew}_{2n}(\mathbb{F}_{q}):\textrm{Pff}(B)\equiv
b\bmod q\\}$

given by:

$\varphi(A)=\begin{pmatrix}b&0&...&0\\\ 0&1&...&0\\\
\vdots&\vdots&\vdots&\vdots\\\
0&0&...&1\end{pmatrix}A\begin{pmatrix}b&0&...&0\\\ 0&1&...&0\\\
\vdots&\vdots&\vdots&\vdots\\\ 0&0&...&1\end{pmatrix}.$

Clearly, if $A^{T}=-A$, then $\varphi(A)^{T}=-\varphi(A)$. We also know that

$\textrm{Pff}(A)=\sum_{j=2}^{2n}(-1)^{j}a_{1j}\textrm{Pff}(A_{\hat{1}\hat{j}}),$

where $A_{\hat{i}\hat{j}}$ is the minor created by removing the $i$th and
$j$th row and column. Since this map $\varphi$ results in multiplying the
first row and column both by $b$, we can see that

$\textrm{Pff}(\varphi(A))=\sum_{j=2}^{2n}ba_{1j}\textrm{Pff}(\varphi(A)_{\hat{i}\hat{j}})=b\sum_{j=2}^{2n}a_{1j}\textrm{Pff}(A_{\hat{i}\hat{j}})=b\textrm{Pff}(A).$

So, we do indeed have
$\varphi:\\{X\in\textrm{Skew}_{2n}(\mathbb{F}_{q}):\textrm{Pff}(X)\equiv
1\bmod
q\\}\rightarrow\\{X\in\textrm{Skew}_{2n}(\mathbb{F}_{q}):\textrm{Pff}(X)\equiv
b\bmod q\\}.$ Since $\text{diag}(b,1,1,...,1)$ is invertible, $\varphi$ is a
bijection between the two sets, so we get that if $(q,b)=1$, the local
equidistribution property (5) holds. ∎

Now, we have all the ingredients necessary to prove Theorem 3.4.

###### Proof of Theorem 3.4.

From Theorem 3.5, for $T$ sufficiently large, we know that

$\displaystyle\pi_{\textrm{Skew}_{2n},\textrm{Pff}}(T\Omega)$
$\displaystyle=(1+o_{\Omega}(1))\mu_{\infty}(\Omega)\sum_{p\leq
T^{n}}p^{2n-2}\cdot\mathfrak{S}_{p}$
$\displaystyle=(1+o_{\Omega}(1))\mu_{\infty}(\Omega)\mathfrak{S}_{\textrm{Skew}_{2n},\textrm{Pff}}\sum_{p\leq
T^{n}}p^{2n-2}(1+O(p^{-1/2}))$
$\displaystyle=(1+o_{\Omega}(1))\mu_{\infty}(\Omega)\mathfrak{S}_{\textrm{Skew}_{2n},\textrm{Pff}}\cdot\frac{T^{n(2n-1)}}{n(2n-1)\log(T)}$
$\displaystyle=(1+o_{\Omega}(1))\mathfrak{S}_{\textrm{Skew}_{2n},\textrm{Pff}}\int_{T\Omega}\frac{d\mathbf{x}}{\log^{+}(\textrm{Pff}(\mathbf{x}))}.$

The second equality follows from Lemma 3.6 and the fourth equality comes from
Lemma 2.5. We note that $\mathfrak{S}_{\textrm{Skew}_{2n},\textrm{Pff}}$ is
the singular series given by the Bateman-Horn conjecture. ∎

Finally, we evaluate the singular series:

$\mathfrak{S}_{\textrm{Skew}_{2n},\textrm{Pff}}=\prod_{p}\frac{\\#\\{X\in\textrm{Skew}_{2n}(\mathbb{F}_{p}):\det(X)\in\mathbb{F}_{p}^{\times}\\}}{p^{n(2n-1)}(1-1/p)}.$

Applying the result of MacWilliams in [26],

(6)
$\\#\\{X\in\textrm{Skew}_{2n}(\mathbb{F}_{p}):\det(X)\in\mathbb{F}_{p}^{\times}\\}=\prod_{i=0}^{n-1}(p^{2n-1}-p^{2i}).$

Thus, we get that

$\mathfrak{S}_{\textrm{Skew}_{2n},\textrm{Pff}}=\prod_{p}(1-1/p)^{-1}\prod_{i=0}^{n-1}(1-p^{1+2i-2n})=\prod_{\begin{subarray}{c}3\leq
j\leq 2n-1\\\ j\textrm{ odd}\end{subarray}}\zeta(j)^{-1}.$

##  4\. $\epsilon$-cuttings

In this section, we will no longer require the dynamical set up of §2. We
assume that $V$ is a linear vector space of dimension larger than one and $F$
is a homogeneous polynomial that is irreducible over $\overline{\mathbb{Q}}$.
Again, we denote our level sets:

$V_{m}=\\{v\in V:F(v)=m\\}.$

We note that since $F$ is homogeneous, we know that

$V_{m}=m^{1/\deg(F)}V_{1}.$

Our goal for the following two sections will be to prove the following
theorem:

######  Theorem 4.1.

Assume that for any nice compact $\Omega\subset V_{1}(\mathbb{R})$, as
$T\rightarrow\infty$, we have that

$\pi_{V,F}(T\Omega)=(1+o_{\Omega}(1))\mathfrak{S}_{V,F}\cdot\int_{T\Omega}\frac{1}{\log^{+}(F(x))}dx.$

If a compact set $\mathcal{R}_{0}\subset V(\mathbb{R})$ has the following
properties:

  1. (1)

If $x\in\mathcal{R}_{0}$ and $t\in[0,1]$, then $tx\in\mathcal{R}_{0},$

  2. (2)

The compact set $[0,\infty)\mathcal{R}_{0}\cap V_{1}$ is nice,

then we have that as $T\rightarrow\infty$,

$\pi_{V,F}(T\mathcal{R}_{0})=(1+o(1))\mathfrak{S}_{V,F}\cdot\int_{T\mathcal{R}_{0}}\frac{1}{\log^{+}(F(x))}dx.$

Moreover, if we also have that for any nice compact $\Omega\subset
V_{-1}(\mathbb{R})$, as $T\rightarrow\infty$, the following holds:

$\\#\\{x\in T\Omega:-F(v)\text{ is
prime}\\}=(1+o_{\Omega}(1))\mathfrak{S}_{V,F}\int_{T\Omega}\frac{1}{\log^{+}(|F(\mathbf{x})|)}dx,$

then Conjecture 1.3 holds for the compact regions $\mathcal{R}_{0}\subset
V(\mathbb{R})$.

######  Remark 4.2.

We note that the Euclidean box $\mathcal{B}=[0,1]^{n}\subset\mathbb{R}^{n}$
would satisfy the conditions $(1),(2)$ above; hence, a consequence of Theorem
4.1 is that under the assumptions given, Conjecture 1.2 holds.

First, we will approximate the region $T\mathcal{R}_{0}$ with such cones
$T\Omega$ using $\epsilon$-cuttings.

######  Definition 4.3.

For $\Omega\subset V_{1}(\mathbb{R})$, a nice compact subset, we define the
height of the subset as

$0pt(\Omega)=\sup_{A\in\Omega}\|A\|,$

where $\|.\|$ is the $L^{\infty}$ norm on $V(\mathbb{R})$.

######  Definition 4.4.

An $\epsilon$-cutting of $\mathcal{R}_{0}$ is a finite set of disjoint nice
connected compact subsets of $V_{1}(\mathbb{R})$, denoted as

$\mathcal{C}_{\epsilon}=\\{\Omega_{i}\\}_{i=1}^{N(\epsilon)},$

such that

$\mathcal{R}_{0}=\mathcal{E}\bigcup_{\Omega\in\mathcal{C}_{\epsilon}}[0,1/0pt(\Omega)]\Omega,$

where the exceptional set satisfies that $|\mathcal{E}|\leq\epsilon.$

######  Remark 4.5.

We note that such an $\epsilon$-cutting of $\mathcal{R}_{0}$ allows us to cut
up $T\mathcal{R}_{0}$ for any $T\geq 1$; we have that

$T\mathcal{R}_{0}=T\mathcal{E}\cup_{\Omega\in
C_{\epsilon}}[0,T/0pt(\Omega)]\Omega,$

and that $|T\mathcal{E}|\leq\epsilon T^{\dim(V)}.$

From now on we will be assuming that the compact set $\mathcal{R}_{0}$ has the
following properties (as in Theorem 4.1):

  1. (1)

If $x\in\mathcal{R}_{0}$ and $t\in[0,1]$, then $tx\in\mathcal{R}_{0}.$

  2. (2)

The compact set $[0,\infty)\mathcal{R}_{0}\cap V_{1}$ is nice.

Next, we show the existence of an $\epsilon$-cutting.

######  Proposition 4.6.

Fix an $\epsilon>0$ sufficiently small. Let $\mathcal{R}_{0}\subset
V(\mathbb{R})$ satisfy assumptions (1) and (2) above. Then, there exists an
$\epsilon$-cutting of $\mathcal{R}_{0}\subset V(\mathbb{R})$.

###### Proof.

Let $L=\mathcal{R}_{0}\cap\\{v\in V(\mathbb{R}):F(v)=0\\}.$ Since
$\mathcal{R}_{0}$ is compact, $L$ is a compact set. Moreover,
$\textrm{vol}(L)=0$, since $L$ is contained in a subvariety of codimension
one. Hence, we can find a sufficiently small $\delta$, such that the set

$L_{\delta}=\\{v\in\mathcal{R}_{0}:\textrm{dist}(v,L)<\delta\\}$

has volume less than $\epsilon/2$. We set $\mathcal{E}_{0}=L_{\delta}.$

Let us consider the set $S=\mathcal{R}_{0}-L_{\delta}$ and split $S$ into its
connected components and such that each component is contained in a separate
orbit of the action of $G(\mathbb{R})$ on $V(\mathbb{R})$, which will produce
a finite number $k_{V}$ of $S=\cup_{i\leq k_{V}}S_{i}$, where $k_{V}$ is
independent of $\epsilon$. We will abuse notation and denote one of these
components as $S$. Now, we wish to split $S$ up further.

We consider

$S_{1}=\mathbb{R}_{+}S\cap V_{1}(\mathbb{R})$

and claim that $S_{1}$ is a compact set. Indeed, since $S$ is a compact set
disjoint from the zero locus of $F$, we know that

$m_{0}(\epsilon)=\min_{u\in S}\left|F(u)\right|>0.$

Thus, $S_{1}\subset V_{1}(\mathbb{R})\cap[0,1/m_{0}(\epsilon)^{1/\deg(F)}]S$
and hence it is compact. So, we have that

$0<m(\epsilon)=\inf_{v\in S_{1}}\|v\|\leq\sup_{v\in
S_{1}}\|v\|=M(\epsilon)<\infty.$

Let $\epsilon^{\prime}$ be a function of $\epsilon$ that we will determine
later. Let us split the interval $[m(\epsilon),M(\epsilon)]$ into intervals of
length at most $\eta(\epsilon)=\epsilon^{\prime}\min(1,M(\epsilon)^{-1})$ and
enumerate these intervals as $I_{k}=[m_{k}-\eta(\epsilon),m_{k}]$ (We can
assume that $M(\epsilon)-m(\epsilon)$ is an integer multiple of
$\eta(\epsilon)$, after passing to a possibly smaller constant). Then we set

$S_{1,k}=\\{v\in S_{1}:\|v\|\in I_{k}\\}.$

and finally

$S^{\prime}=\bigcup_{k}[0,1/m_{k}]S_{1,k}.$

Note that $S^{\prime}\subset
S\subset\cup_{k}[0,1/(m_{k}-\eta(\epsilon))]S_{1,k}$. Observe now that

$\textrm{vol}([1/m_{k},1/(m_{k}-\eta(\epsilon))]S_{1,k})\leq(m_{k}-\eta(\epsilon))^{-\dim(V)}-m_{k}^{-\dim(V)}\leq
c_{V}\eta(\epsilon)m_{k}^{-\dim(V)},$

where $c_{V}$ is a constant in terms of $\dim(V)$. So, we know that

$\sum_{k}\eta(\epsilon)m_{k}^{-\dim(V)}\leq
C\eta(\epsilon)\int_{m(\epsilon)}^{M(\epsilon)}m^{-\dim(V)}dm\leq
C_{V}^{\prime}\eta(\epsilon),$

where $C,C_{V}^{\prime}>0$ are constants. Thus, we have that by squeezing,

$\textrm{vol}(S-S^{\prime})\leq c_{V}C_{V}^{\prime}\epsilon^{\prime}.$

We choose $\epsilon^{\prime}\leq\frac{\epsilon}{2c_{V}C_{V}^{\prime}}$ and set
$\mathcal{E}=(S-S^{\prime})\cup\mathcal{E}_{0}.$ For such a choice, we have
that

$\mathcal{R}_{0}=\mathcal{E}\cup\bigcup_{i}\bigcup_{k}[1,0pt(S_{i,k})]S_{i,k},$

with $\textrm{vol}(\mathcal{E})\leq\epsilon.$ ∎

##  5\. An upper bound sieve

In this section, we will first establish a general upper bound for the number
of prime values of an irreducible multivariate polynomial; this upper bound
will be of the same size as what the Bateman-Horn conjecture (Conjecture 1.2)
predicts. We note that we do not need the set up of the general framework for
this bound.

Let $F(\mathbf{x})\in\mathbb{Z}[\mathbf{x}]$ be an irreducible polynomial in
$n\geq 2$ variables. Let us denote the box of height $T$ by

$B(T):=\\{\mathbf{x}\in\mathbb{R}^{n}:\|\mathbf{x}\|_{\infty}\leq T\\}.$

Let us also write $\mathscr{P}$ as the set of primes. Then we get the
following upper bound for the number of prime values of $F(\mathbf{x})$:

######  Proposition 5.1.

Let $\mathcal{R}\subset B(T)$ be a convex region. Then there is a constant
$c_{F}>0$ such that

$\\#\\{\mathbf{x}\in\mathcal{R}(\mathbb{Z}):F(\mathbf{x})\in\mathscr{P}\\}\leq
c_{F}\left(\frac{|\mathcal{R}|}{\log(T)}+T^{n-1/2}\right).$

###### Proof.

Let $z=T^{\frac{1}{10n}}$ and we write

$P(z)=\prod_{\begin{subarray}{c}p\in\mathscr{P}\\\ p\leq z\end{subarray}}p.$

We can bound the number of prime values of $F$ in $\mathcal{R}$ using the
$z$-rough numbers as follows:

$\\#\\{\mathbf{x}\in\mathcal{R}(\mathbb{Z}):F(\mathbf{x})\in\mathscr{P}\\}\leq\\#\\{\mathbf{x}\in\mathcal{R}(\mathbb{Z}):F(\mathbf{x})\in\mathscr{P}\cap[1,z]\\}+\\#\\{\mathbf{x}\in\mathcal{R}(\mathbb{Z}):\gcd(F(\mathbf{x}),P(z))=1\\},$

First we apply Lemma 6.8 of Opera de Cribro [17] to the count of $z$-rough
numbers. We then know that for a fixed constant $C>0$, the following
inequality holds:

(7)
$\\#\\{\mathbf{x}\in\mathcal{R}(\mathbb{Z}):\gcd(F(\mathbf{x}),P(z))=1\\}\leq
C|\mathcal{R}|\prod_{p\leq
z}\left(1-\frac{\\#\\{\mathbf{x}\in\mathbb{F}_{p}^{n}:F(\mathbf{x})\equiv
0\bmod p\\}}{p^{n}}\right)\\\ +O\left(\sum_{d\leq
z}\mu^{2}(d)|r_{d}(\mathcal{R})|\right),$

where the $r_{d}(\mathcal{R})$ is the remainder term:

$r_{d}(\mathcal{R})=\\#\\{\mathbf{x}\in\mathcal{R}(\mathbb{Z}):F(\mathbf{x})\equiv
0\bmod
d\\}-|\mathcal{R}|\cdot\frac{\\#\\{\mathbf{x}\in(\mathbb{Z}/d\mathbb{Z})^{n}:F(\mathbf{x})\equiv
0\bmod d\\}}{d^{n}}.$

First, we consider the product over primes $p\leq z$:

$\displaystyle\prod_{p\leq
z}\left(1-\frac{\\#\\{\mathbf{x}\in\mathbb{F}_{p}^{n}:F(\mathbf{x})\equiv
0\bmod p\\}}{p^{n}}\right)$ $\displaystyle\ll_{F}\exp(-\sum_{p\leq
z}\frac{\\#\\{\mathbf{x}\in\mathbb{F}_{p}^{n}:F(\mathbf{x})\equiv 0\bmod
p\\}}{p^{n}}+O(p^{-2}))$ $\displaystyle\ll_{F}\exp(-\sum_{p\leq
z}p^{-1}+O(p^{-3/2}))$ $\displaystyle\ll_{F}\log(z)^{-1}.$

In the first inequality, we use that

$\varrho(p):=\\#\\{\mathbf{x}\in\mathbb{F}_{p}^{n}:F(\mathbf{x})\equiv 0\bmod
p\\}\leq dp^{n-1},$

so we know that

$\log(1-\frac{\varrho(p)}{p^{n}})=-\frac{\varrho(p)}{p^{n}}+O(p^{-2}).$

For the second inequality, we use that since $F$ is absolutely irreducible, we
have an even better estimate for
$\\#\\{\mathbf{x}\in\mathbb{F}_{p}^{n}:F(\mathbf{x})\equiv 0\bmod p\\}$ for
$p$ sufficiently large in terms of $F$:

$\\#\\{\mathbf{x}\in\mathbb{F}_{p}^{n}:F(\mathbf{x})\equiv 0\bmod
p\\}=p^{n-1}+O(p^{n-3/2}).$

Moreover, for any $\alpha>1$, we know that

$\sum_{p\leq z}p^{-\alpha}=O(1),$

whereas for $\alpha=1$, we have

$\sum_{p\leq z}p^{-1}=\log\log(z)+O(1).$

Together these give the final inequality.

Next, we need a nontrivial level of distribution result:

$\sum_{d\leq T^{1/10n}}\mu^{2}(d)|r_{d}(\mathcal{R})|\ll_{F}T^{n-1/2}.$

For a fixed $d$, we can write

(8) $r_{d}(\mathcal{R})=\sum_{\begin{subarray}{c}\alpha\bmod d\\\
F(\alpha)\equiv 0\bmod
d\end{subarray}}\left(\sum_{\begin{subarray}{c}\mathbf{x}\in\mathcal{R}(\mathbb{Z})\\\
v\equiv\alpha\bmod d\end{subarray}}1-\frac{|\mathcal{R}|}{d^{n}}\right).$

We estimate the internal sum using geometry of numbers. We remark that if we
define

$\Lambda(\alpha,d):=\\{\mathbf{x}\in\mathbb{Z}^{n}:\mathbf{x}\equiv\alpha\bmod
d\\},$

then $\Lambda(\alpha,d)$ is a lattice of determinant $d^{n}.$ Now, since
$\mathcal{R}\subset B(T)$, we know that the longest “sidelength” of the region
is $\ll T$. Moreover, it is a convex region and thus, we get that

$\\#\\{\mathbf{x}\in\mathcal{R}(\mathbb{Z})\cap\Lambda(\alpha,d)\\}=\frac{|\mathcal{R}|}{d^{n}}+O\left(\max_{1\leq
k\leq n}\frac{T^{n-k}}{\lambda_{1}\ldots\lambda_{n-k}}\right),$

where $\lambda_{1},...,\lambda_{n}$ are the successive minima of the lattice
$\Lambda(\alpha,d).$ Since $\lambda_{i}\geq 1$ for each $i$, we use the
trivial bound that

$\\#\\{v\in\mathcal{R}(\mathbb{Z})\cap\Lambda(\alpha,d)\\}-\frac{|\mathcal{R}|}{d^{n}}\ll
T^{n-1}.$

Thus, we have that (8) is bounded by

$|r_{d}(\mathcal{R})|\leq\sum_{\begin{subarray}{c}\alpha\bmod d\\\
F(\alpha)\equiv 0\bmod d\end{subarray}}T^{n-1}\ll T^{n-1}d^{n-1}.$

Here we have used that $F(v)\equiv 0\bmod d$ cuts out a subvariety of
codimension $1$. So, we can calculate that

$\sum_{d\leq T^{1/10n}}\mu^{2}(d)|r_{d}(\mathcal{R})|\ll_{F}T^{n-9/10}\leq
T^{n-1/2}.$

Plugging the above bound into (7), we achieve:

(9)
$\\#\\{\mathbf{x}\in\mathcal{R}(\mathbb{Z}):\gcd(F(\mathbf{x}),P(z))=1\\}\ll_{F}\frac{|\mathcal{R}|}{\log(T)}+T^{n-1/2}.$

Finally, we need to bound the contribution from the small primes less than
$z$:

$\displaystyle\\#\\{\mathbf{x}\in\mathcal{R}(\mathbb{Z}):F(\mathbf{x})\in\mathscr{P}\cap[1,T^{1/10n}]\\}$
$\displaystyle\leq\sum_{m=1}^{T^{1/10n}}\\#\\{\mathbf{x}\in\mathcal{R}(\mathbb{Z}):F(\mathbf{x})=m\\}$
$\displaystyle\leq\sum_{m=1}^{T^{1/10n}}\\#\\{\mathbf{x}\in
B(T)(\mathbb{Z}):F(\mathbf{x})=m\\}.$

Let us write $\mathbf{x}=(x_{1},...,x_{n})$ where we assume that
$F(\mathbf{x})=G(x_{1},...,x_{n-1},x_{n})x_{n}+H(x_{1},...,x_{n-1})$ for
$G(x_{1},...,x_{n-1},x_{n})$ nontrivial. Then we can write the sum above as:

$\leq\sum_{|x_{1}|\leq T}\ldots\sum_{|x_{n-1}|\leq
T}\sum_{m=1}^{T^{1/10n}}\\#\\{|x_{n}|\leq T:F(x_{1},...,x_{n-1},x_{n})=m\\}.$

Now if $F(x_{1},...,x_{n-1},x_{n})$ as a polynomial in $x_{n}$ has degree
$d\geq 1$, we have that

$\\#\\{|x_{n}|\leq T:F(x_{1},...,x_{n})=m\\}\leq d$

for any value of $d$. So, these values of $x_{1},...,x_{n-1}$ will contribute
to an upper bound of $T^{n-1+1/10n}$ values.

On the other hand, if $F(x_{1},...,x_{n-1},x_{n})=c$ is a constant polynomial
in $x_{n}$, for $m=c$, we get that

$\\#\\{|x_{n}|\leq T:F(x_{1},...,x_{n-1},x_{n})=c\\}\leq T$

and otherwise

$\\#\\{|x_{n}|\leq T:F(x_{1},...,x_{n-1},x_{n})=m\\}=0.$

However, for $F(x_{1},...,x_{n-1},x_{n})$ to be a constant polynomial,
$x_{1},...,x_{n-1}$ must satisfy $\tilde{G}(x_{1},...,x_{n-1})=0$ where
$\tilde{G}(x_{1},...,x_{n-1})$ is the coefficient of any positive power of
$x_{n}$ in $F(x_{1},...,x_{n-1},x_{n})$. We know that

$\\#\\{|x_{i}|\leq T:\tilde{G}(x_{1},...,x_{n-1})=0\\}\ll_{\tilde{G}}T^{n-2},$

since $\tilde{G}$ cuts out a codimension one variety. So, combining this with
the above bound, we get that

$\\#\\{\mathbf{x}\in\mathcal{R}(\mathbb{Z}):F(\mathbf{x})\in\mathscr{P}\cap[1,T^{1/10n}]\\}\leq\sum_{m=1}^{T^{1/10n}}\\#\\{\mathbf{x}\in
B(T)(\mathbb{Z}):F(\mathbf{x})=m\\}\ll_{F}T^{n-1+1/10n}.$

Combining the above bound with (9), we achieve final upper bound.

∎

We remark that for $F$ a homogeneous polynomial, this level of distribution
result can be improved by studying the level of distribution of lattices, as
done by Daniel in [9], or by using bounds on exponential sums, as done by
Friedlander and Iwaniec in Chapter 22 of [17].

###  5.1. Proof of Theorem 4.1

From Proposition 4.6, we know that we have an $\epsilon$-cutting of
$\mathcal{R}_{0}$. Thus if we define

$\pi_{V,F}(T\mathcal{R}_{0}):=\\#\\{v\in V(\mathbb{Z})\cap
T\mathcal{R}_{0}:F(v)\textrm{ is prime}\\},$

then we have that for every $T>1$:

$\pi_{V,F}(T\mathcal{R}_{0})=\sum_{\Omega\in\mathcal{C}_{\epsilon}}\pi_{V,F}(T/0pt(\Omega)\Omega))+\\#\\{v\in
V(\mathbb{Z})\cap T\mathcal{E}:F(v)\textrm{ is prime}\\}.$

First we handle the error term with a sieve argument. We note that in the
construction of the $\epsilon-$cutting from Proposition 4.6, $\mathcal{E}$
will be the finite union of convex regions contained in $\mathcal{R}_{0}$.
Then applying Proposition 5.1 on each of the convex regions, we get that there
is a constant $C_{F,V}\geq 0$ such that

$\\#\\{v\in V(\mathbb{Z})\cap T\mathcal{E}:F(v)\text{ is prime}\\}\leq
C_{F,V}\left(\frac{|T\mathcal{E}|}{\log(T)}+T^{\dim(V)-1/2}\right)\leq
C_{F,V}\frac{\epsilon T^{\dim(V)}}{\log(T)}.$

Now, we handle our main term. Under our assumptions, we have that for
$\Omega\subset V_{1}(\mathbb{R})$ a nice compact subset,

$\pi_{V,F}(T/0pt(\Omega)\Omega)=(1+o_{\Omega}(1))\mathfrak{S}_{V,F}\int_{T/0pt(\Omega)\Omega}\frac{1}{\log^{+}(F(x))}dx.$

Thus, our main term becomes

$\sum_{\Omega\in\mathcal{C}_{\epsilon}}\pi_{V,F}(T/0pt(\Omega)\Omega)=(1+o_{\epsilon}(1))\mathfrak{S}_{V,F}\int_{\cup_{\mathcal{C}_{\epsilon}}T/0pt(\Omega)\Omega}\frac{1}{\log^{+}(F(x))}dx.$

By taking $\epsilon\rightarrow 0$, we obtain

$\pi_{V,F}(T\mathcal{R}_{0})=(1+o(1))\mathfrak{S}_{V,F}\int_{T\mathcal{R}_{0}}\frac{1}{\log^{+}(F(x))}dx.$

By the same argument, if the second assumption holds in Theorem 4.1, then by
using an $\epsilon$-cutting of $\mathcal{R}_{0}$ with nice compact subsets
$\Omega\subset V_{-1}(\mathbb{R})$, the sum over negative primes achieves the
final statement of the theorem. ∎

###  5.2. Proof of Theorem 1.4 and Theorem 1.6

In this section, we combine Theorem 3.1 and Theorem 3.4 with Theorem 4.1 to
achieve Theorem 1.4 and Theorem 1.6.

###### Proof of Theorem 1.4.

First, we note that Theorem 3.1 gives us the first assumption of Theorem 4.1.
Next, we remark that the box $\\{\|X\|\leq
T\\}\subset\textrm{Mat}_{n}(\mathbb{R})$ satisfies the conditions on
$\mathcal{R}_{0}$ set by Theorem 4.1. Hence, we get the following result:

$\pi_{\det}(T)=(1+o(1))\cdot\prod_{j=2}^{n}\zeta(j)^{-1}\int_{\|X\|\leq
T}\frac{1}{\log^{+}(\det(X))}dX.$

∎

###### Proof of Theorem 1.6.

First, we note that Theorem 3.4 gives us the first assumption of Theorem 4.1.
Next, we remark that the box $\\{\|X\|\leq
T\\}\subset\textrm{Skew}_{2n}(\mathbb{R})$ satisfies the conditions on
$\mathcal{R}_{0}$ set by Theorem 4.1. Hence, we get the following result:

$\pi_{\textrm{Pff}}(T)=(1+o(1))\cdot\prod_{\begin{subarray}{c}3\leq n\leq
2n-1\\\ j\text{
odd}\end{subarray}}\zeta(j)^{-1}\cdot\int_{\begin{subarray}{c}\|X\|\leq T\\\
X^{T}=-X\end{subarray}}\frac{1}{\log^{+}(\textrm{Pff}(X))}dX.$

∎

##  6\. Background on symmetric matrices and the Siegel Mass Formula

Finally, we handle the more involved example of the determinant polynomial on
symmetric matrices. Let us first discuss how this case does and does not fit
into the general framework outlined in §2.

Define $G=SL_{n}(\mathbb{R})$ and $V$ to be the space of symmetric matrices:

$V=\\{X\in\textrm{Mat}_{n}:X^{T}=X\\}.$

We will take $F$ to be the determinant polynomial on $V$. Then we have that
$g\in G=\textrm{SL}_{n}$ acts on
$V_{1}=\\{X\in\textrm{Mat}_{n}:X^{T}=X,\det(X)=1\\}$ via

$X\mapsto g^{T}Xg.$

We can then see that $V_{1}(\mathbb{R})$ is the finite union of $n+1$ orbits,
where the orbits are parameterized by the signatures of the symmetric
matrices. For $p+q=n$, let $\mathcal{O}_{p,q}$ be the orbit of matrices with
signature $(p,q).$ In this case, the conditions to apply Theorem 1.9 of Oh
[28] are not always satisfied – in particular for the orbit of positive (or
negative) definite symmetric matrices, the stabilizer of the generator
$\text{diag}(1,1,...,1)$ is $\textrm{SO}_{n}(\mathbb{R})$ and thus compact. To
resolve this issue, we use the Linnik equidistribution result of Einsiedler,
Margulis, Mohammadi, and Venkatesh in [13] to get asymptotics for

$\\{X\in\textrm{GL}_{n}(\mathbb{Z})\cap m^{\frac{1}{n}}\Omega:\text{ $X$ is
symmetric positive definite}\\},$

for $\Omega$ some nice compact subset of $\textrm{SL}_{n}(\mathbb{R}).$ To do
so, we use the Siegel mass formula to evaluate the corresponding class
numbers. Moreover, by evaluating the local factors of the Siegel mass formula
we calculate the singular series.

Before proving the prime number theorem for the symmetric case, it is prudent
to review some important facts about the Siegel mass formula. Recall that two
integral positive definite symmetric matrices are in the same genus if they
are equivalent over $\mathbb{Z}_{p}$ for all $p$ and over $\mathbb{R}$. There
are finitely many equivalence classes in each genus. For a positive definite
symmetric matrix $S$ of genus $\mathcal{G}$, the mass $M(S)$ of $S$ is defined
as

$M(S):=\sum_{S_{k}\in\mathcal{G}}\frac{1}{\left|\textrm{Aut}(S_{k})\right|},$

where the sum is over representatives $S_{k}$ in the genus $\mathcal{G}$ of
$S$ and $\textrm{Aut}(S_{k})$ is the automorphism group of $S_{k}$ in
$\textrm{GL}_{n}(\mathbb{Z})$. We note that $M(S)$ only depends on the genus
of $S$, and could be denoted as $M(\mathcal{G})$. Siegel’s celebrated mass
formula evaluates this quantity as a product of local densities.

######  Theorem 6.1 (Siegel [32]).

Let $n\geq 3$ and $S$ be a positive definite symmetric $n\times n$ matrix of
determinant $D$. Then we have the following evaluation of $M(S):$

$M(S)=2\pi^{-n(n+1)/4}\prod_{k=1}^{n}\Gamma(k/2)\prod_{p}\alpha_{p}(S)^{-1}\cdot
D^{(n+1)/2},$

where $\alpha_{p}$ is the $p$-adic density at the prime $p$ defined as

$\alpha_{p}(S):=\frac{1}{2}\lim_{\ell\rightarrow\infty}p^{-\ell
n(n-1)/2}\\#\\{T\in\textrm{Mat}_{n}(\mathbb{Z}/p^{\ell}\mathbb{Z}):T^{T}ST\equiv
S\bmod p^{\ell}\\}.$

For all but finitely many primes $p$ in the Siegel mass formula (for $p\nmid
2D$), the formula for $\alpha_{p}(S)$ has the following clean expression:

(10)
$\alpha_{p}^{\textrm{std}}(S)=\begin{cases}\prod_{k=1}^{(n-1)/2}(1-p^{-2k}),&n\text{
odd},\\\
\prod_{i=1}^{n/2-1}(1-p^{-2k})\cdot(1-p^{-n/2}\genfrac{(}{)}{}{}{(-1)^{n/2}D}{p}),&n\text{
even.}\end{cases}$

We will call this the standard $p$-density. Here the Legendre symbol
$\genfrac{(}{)}{}{}{(-1)^{n/2}D}{p}$ is interpreted as $0$ if $p\mid 2D.$

Evaluating $\alpha_{p}(S)$ at the nonstandard primes can be tricky;
calculating the local factors of the Siegel mass formula for $p=2$ is a
notorious problem. Fortunately, we will only need the case when the
determinant $D$ is a prime and hence the only small divisor we have to deal
with is $2$. We note that both Kitaoka [21] and Kovaleva [22] evaluated these
densities at odd primes; additionally, Kitaoka evaluates $\alpha_{p}(S)$ for
all primes when $n$ is odd.

###  6.1. The density at nonstandard primes $p\neq 2$

Let $p$ be a nonstandard prime, we want to look at the $p$-adic density
$\alpha_{p}(S)$. Let us assume that over $\mathbb{Q}_{p}$, $S$ has the Jordan
decomposition

$S=\sum_{k=-\infty}^{\infty}p^{k}f_{k},$

where $f_{k}$ is the Jordan block with determinant coprime to $p$. We note
that $f_{k}=0$ for all but finitely many values of $k$. Let us denote the
determinant of $f_{k}$ as $\Delta_{k}$ and the dimension of the Jordan block
as $n_{k}$. Then for $p\neq 2$, the formula for the $p$-adic density is given
by Conway and Sloane in [8] as the following:

(11)
$\alpha_{p}(S)^{-1}=2p^{-\frac{s(n+1)}{2}}\prod_{k}M_{p}(f_{k})\prod_{k^{\prime}>k}p^{\frac{(k^{\prime}-k)n_{k}n_{k^{\prime}}}{2}}$

where $p^{s}\|D$, the products over $k$ and $k^{\prime}$ are over the indices
such that $f_{k}\neq 0$, and

$M_{p}(f_{k})=\begin{cases}\frac{1}{2}\prod_{i=1}^{\frac{n-1}{2}}(1-p^{-2i})^{-1},&n\text{
is odd},\\\
\frac{1}{2}(1-\genfrac{(}{)}{}{}{(-1)^{n/2}\Delta_{k}}{p}p^{-n/2})^{-1}\cdot\prod_{i=1}^{\frac{n-2}{2}}(1-p^{-2i})^{-1},&n\text{
is even.}\end{cases}$

Notice that this formula still holds for $p$ a standard prime and hence are
definitions are consistent.

######  Lemma 6.2.

Let $p$ be an odd prime and $n\geq 3$, then for $S$ a positive definite
symmetric matrix of determinant $p$, we have that

$\alpha_{p}(S)^{-1}=\frac{p^{-1}}{2}\cdot\alpha_{p}^{\textrm{std}}(S)^{-1}\cdot\left(1+O(p^{-2})\right).$

###### Proof.

Since $p$ is an odd prime, we have that the only nontrivial Jordan blocks are
$f_{0}$ and $f_{1}$. Moreover, $f_{0}$ has dimension $n_{0}=n-1$ and $f_{1}$
has dimension $n_{1}=1$. Thus, we get that

$\alpha_{p}(S)^{-1}=2p^{-1}M_{p}(f_{0})M_{p}(f_{1}).$

Evaluating $M_{p}(f_{0})$ and $M_{p}(f_{1})$, we get that

$\alpha_{p}(S)^{-1}=\frac{p^{-1}}{2}\cdot\alpha_{p}^{\textrm{std}}(S)^{-1}\cdot\left(1+O(p^{-2})\right).$

∎

###  6.2. The $2$-adic density

For the $2$-adic density, there is a more complicated formula than (11) that
takes into account the specifics of the Jordan decomposition.

######  Definition 6.3.

A Jordan block $f_{k}$ is said to be Type I if $f_{k}$ represents an odd
number in $\mathbb{Z}_{2}$. In this case, $f_{k}$ can be taken to be diagonal.
Otherwise, $f_{k}$ is said to be Type II and will be a direct sum of matrices
of the form

$\left\\{\begin{pmatrix}2w&u\\\
u&2v\end{pmatrix}:w,u,v\in\mathbb{Z}_{2}\right\\}.$

We note that Type II Jordan blocks can only occur if $n$ is even.

In [8], Conway and Sloane introduce the notion of love forms, which are
$0$-dimensional Jordan blocks that still contribute to the formula.

######  Definition 6.4.

A Jordan block $f_{k}$ with $n_{k}=0$ is called a love form. These forms are
considered Type II forms.

######  Remark 6.5.

We note that in some sense love forms are nonexistent, as they are zero-
dimensional Jordan blocks. Nonetheless, they are a useful trick and convention
to simplify the exposition around the Siegel mass formula at $p=2$.

If $f_{k}$ is a love form and $f_{k+1}$ or $f_{k-1}$ is a Type I Jordan block,
then there is a contribution of $M_{2}(f_{k})=\frac{1}{2}$; such a form
$f_{k}$ is called a bound love form. Otherwise, $f_{k}$ is called a free love
form, and $M_{2}(f_{k})=1$. With this context, we can introduce the
modification of (11) at the place $p=2$:

(12) $\alpha_{p}(S)^{-1}=2\cdot
2^{-\frac{s(n+1)}{2}}\prod_{k}M_{2}(f_{k})\prod_{k^{\prime}>k}2^{\frac{(k^{\prime}-k)n_{k}n_{k^{\prime}}}{2}}\times
2^{n(I,I)-n(II)},$

where $2^{s}\|D$, $n(I,I)$ is the number of adjacent Type I Jordan blocks, and
$n(II)$ is the sum of the dimensions of all of the Jordan blocks of Type II.

In our case, since $D$ is some odd prime, the only relevant forms are $f_{0}$
and the love forms. If $f_{0}$ is a nontrivial Jordan block, the value of
$M_{2}(f_{0})$ depends on a$\bmod 8$ invariant.

######  Definition 6.6.

Let $S$ denote some positive definite integral matrix of discriminant $D$. If
$f_{0}$ is Type I, then it is diagonalizable and we take $E_{i}$ to be the
number of elements on the diagonal that are congruent to $i\bmod 4$. We define
the octane $\mathcal{O}_{S}$ to be:

$\mathcal{O}_{S}:=\begin{cases}E_{1}-E_{3},&f_{1}\text{ is Type I},\\\
4-4\genfrac{(}{)}{}{}{D}{2},&f_{1}\text{ is Type II}.\end{cases}$

The octane is a$\bmod 8$ invariant.

Now the definition of $M_{2}(f_{0})$ is given by the following:

(13)
$M_{2}(f_{0})=\frac{1}{2}\times\begin{cases}\prod_{i=1}^{\frac{n-3}{2}}(1-2^{-2i})^{-1}\cdot(1-2^{\frac{1-n}{2}})^{-1},&n\text{
is odd, and $\mathcal{O}_{S}\equiv\pm 1\bmod 8$,}\\\
\prod_{i=1}^{\frac{n-3}{2}}(1-2^{-2i})^{-1}\cdot(1+2^{\frac{1-n}{2}})^{-1},&n\text{
is odd, and $\mathcal{O}_{S}\equiv\pm 3\bmod 8$,}\\\
\prod_{i=1}^{\frac{n-2}{2}}(1-2^{-2i})^{-1},&n\text{ is even, $f_{0}$ is Type
I, and $\mathcal{O}_{S}\equiv\pm 2\bmod 8$},\\\
\prod_{i=1}^{\frac{n-4}{2}}(1-2^{-2i})^{-1}\cdot(1-2^{1-\frac{n}{2}})^{-1},&n\text{
is even, $f_{0}$ is Type I, and $\mathcal{O}_{S}\equiv 0\bmod 8$},\\\
\prod_{i=1}^{\frac{n-4}{2}}(1-2^{-2i})^{-1}\cdot(1+2^{1-\frac{n}{2}})^{-1},&n\text{
is even, $f_{0}$ is Type I, and $\mathcal{O}_{S}\equiv 4\bmod 8$},\\\
\prod_{i=1}^{\frac{n-2}{2}}(1-2^{-2i})^{-1}\cdot(1-2^{-\frac{n}{2}})^{-1},&n\text{
is even, $f_{0}$ is Type II, and $\mathcal{O}_{S}\equiv 0\bmod 8$},\\\
\prod_{i=1}^{\frac{n-2}{2}}(1-2^{-2i})^{-1}\cdot(1+2^{-\frac{n}{2}})^{-1},&n\text{
is even, $f_{0}$ is Type II, and $\mathcal{O}_{S}\equiv 4\bmod
8$}.\end{cases}$

Let us evaluate the $2$-density for when $D$ is an odd prime.

######  Lemma 6.7.

Let $D$ be an odd prime and $n\geq 3$. Then for $S$ a positive definite
symmetric matrix of determinant $D$, we have that if $n$ is odd, then

$\alpha_{2}(S)^{-1}=\frac{1}{4}\prod_{i=1}^{\frac{n-3}{2}}(1-2^{-2i})^{-1}\times\begin{cases}(1-2^{\frac{1-n}{2}})^{-1},&\mathcal{O}_{S}\equiv\pm
1\bmod 8,\\\ (1+2^{\frac{1-n}{2}})^{-1},&\mathcal{O}_{S}\equiv\pm 3\bmod
8.\end{cases}$

If $n$ is even, then we have that

$\alpha_{2}(S)^{-1}=\begin{cases}\frac{1}{4}\prod_{i=1}^{\frac{n-2}{2}}(1-2^{-2i})^{-1},&n\text{
is even, $f_{0}$ is Type I, and $\mathcal{O}_{S}\equiv\pm 2\bmod 8$},\\\
\frac{1}{4}\prod_{i=1}^{\frac{n-4}{2}}(1-2^{-2i})^{-1}\cdot(1-2^{1-\frac{n}{2}})^{-1},&n\text{
is even, $f_{0}$ is Type I, and $\mathcal{O}_{S}\equiv 0\bmod 8$},\\\
\frac{1}{4}\prod_{i=1}^{\frac{n-4}{2}}(1-2^{-2i})^{-1}\cdot(1+2^{1-\frac{n}{2}})^{-1},&n\text{
is even, $f_{0}$ is Type I, and $\mathcal{O}_{S}\equiv 4\bmod 8$},\\\
2^{-n}\prod_{i=1}^{\frac{n-2}{2}}(1-2^{-2i})^{-1}\cdot(1-2^{-\frac{n}{2}})^{-1},&n\text{
is even, $f_{0}$ is Type II, and $\mathcal{O}_{S}\equiv 0\bmod 8$},\\\
2^{-n}\prod_{i=1}^{\frac{n-2}{2}}(1-2^{-2i})^{-1}\cdot(1+2^{-\frac{n}{2}})^{-1},&n\text{
is even, $f_{0}$ is Type II, and $\mathcal{O}_{S}\equiv 4\bmod
8$}.\end{cases}$

###### Proof.

We will prove this by looking at the various cases. Since $D$ is an odd prime,
we know the only nontrivial Jordan block occurs for $f_{0}$. First, we assume
$n$ is odd. Then, we have that

$\displaystyle\alpha_{2}(S)^{-1}$
$\displaystyle=2M_{2}(f_{-1})M_{2}(f_{0})M_{2}(f_{1})$
$\displaystyle=\frac{1}{4}\prod_{i=1}^{\frac{n-3}{2}}(1-2^{-2i})^{-1}\times\begin{cases}(1-2^{\frac{1-n}{2}})^{-1},&\mathcal{O}_{S}\equiv\pm
1\bmod 8,\\\ (1+2^{\frac{1-n}{2}})^{-1},&\mathcal{O}_{S}\equiv\pm 3\bmod
8.\end{cases}$

Here we have inputted the contributions from the love forms $f_{-1}$ and
$f_{1}$. If $n$ is even, we have that

$\displaystyle\alpha_{2}(S)^{-1}$
$\displaystyle=2M_{2}(f_{-1})M_{2}(f_{0})M_{2}(f_{1})\cdot 2^{n(I,I)-n(II)}$
$\displaystyle=\begin{cases}\frac{1}{4}\prod_{i=1}^{\frac{n-2}{2}}(1-2^{-2i})^{-1},&n\text{
is even, $f_{0}$ is Type I, and $\mathcal{O}_{S}\equiv\pm 2\bmod 8$},\\\
\frac{1}{4}\prod_{i=1}^{\frac{n-4}{2}}(1-2^{-2i})^{-1}\cdot(1-2^{1-\frac{n}{2}})^{-1},&n\text{
is even, $f_{0}$ is Type I, and $\mathcal{O}_{S}\equiv 0\bmod 8$},\\\
\frac{1}{4}\prod_{i=1}^{\frac{n-4}{2}}(1-2^{-2i})^{-1}\cdot(1+2^{1-\frac{n}{2}})^{-1},&n\text{
is even, $f_{0}$ is Type I, and $\mathcal{O}_{S}\equiv 4\bmod 8$},\\\
2^{-n}\prod_{i=1}^{\frac{n-2}{2}}(1-2^{-2i})^{-1}\cdot(1-2^{-\frac{n}{2}})^{-1},&n\text{
is even, $f_{0}$ is Type II, and $\mathcal{O}_{S}\equiv 0\bmod 8$},\\\
2^{-n}\prod_{i=1}^{\frac{n-2}{2}}(1-2^{-2i})^{-1}\cdot(1+2^{-\frac{n}{2}})^{-1},&n\text{
is even, $f_{0}$ is Type II, and $\mathcal{O}_{S}\equiv 4\bmod
8$}.\end{cases}$

Here we have used that if $f_{0}$ is Type II, we have that
$2^{n(I,I)-n(II)}=2^{-n}$ and no contribution from $f_{-1}$ or $f_{1}$ since
they are both free love forms. In the case, where $f_{0}$ is Type I, we have
used that $2^{n(I,I)-n(II)}=1$ and we have a contribution of $\frac{1}{2}$
from each of the two bound love forms $f_{-1}$ and $f_{1}$. ∎

##  7\. Symmetric matrices

Let $V=\textrm{Sym}_{n}=\\{X\in\textrm{Mat}_{n}:\,X^{T}=X\\}$ be a
$\mathbb{Z}$-linear space. Unlike the case of $(\textrm{Mat}_{n},\det)$, the
pair ($\textrm{Sym}_{n},\det$) is not a Hardy-Littlewood system; however,
along the indefinite orbits it is a “almost” a Hardy-Littlewood system. We
will first treat these orbits.

###  7.1. The indefinite orbit

Let us fix $p,q$ such that $p+q=n$ and $p,q\neq 0.$ For the orbits
$\mathcal{O}_{p,q}$ of symmetric matrices with signature $(p,q)$, we apply the
following result of Oh in [28]. Here we denote by $\mu_{m}$ the Tamagawa
measure on the variety $V_{m}(\mathbb{A}):$

######  Theorem 7.1 (Oh [28], Example 4.3).

Let $n\geq 3$. Let $\Omega\subset\mathcal{O}_{p,q}$ be a nice connected
compact subset. Then we have that as $m\rightarrow\infty$,

$N(m,\Omega)=\\#\\{V_{m}(\mathbb{Z})\cap\mathbb{R}^{+}\Omega\\}=(1+o_{\Omega}(1))\int_{m^{\frac{1}{n}}\Omega\times\prod\limits_{p}V_{m}(\mathbb{Z}_{p})}\delta(x)d\mu_{m}(x).$

Here, the density function $\delta:V_{m}(\mathbb{A})\rightarrow\mathbb{R}$ is
constant on any adelic orbit $\mathcal{O}(\mathbb{A})\subset
V_{m}(\mathbb{A})$ and satisfies that for $x\in\mathcal{O}(\mathbb{A})$ we
have:

$\delta(x)=\begin{cases}2,&\mathcal{O}(\mathbb{A})\text{ contains a
$\mathbb{Q}$-rational point},\\\ 0,&\text{otherwise.}\\\ \end{cases}.$

######  Remark 7.2.

As described in Example 6.3 of [3], we can interpret $x\in V_{m}(\mathbb{A})$
itself as a quadratic form and thus

$\delta(x)\neq 0\iff\prod_{p}c_{p}(x_{p})=1,$

where $c_{p}(x_{p})$ are the Hasse-Minkowski invariants (see [6]) and the
product ranges over both finite and infinite places.

Restricting to the case of when $m$ is some odd prime $q$, we notice that for
any prime $p\neq 2,q,$ there is a unique
$\textrm{SL}_{n}(\mathbb{Z}_{p})-$orbit inside of $V_{q}(\mathbb{Z}_{p}).$
Furthermore, in this case for any $x\in V_{q}(\mathbb{A})$, we have that

$c_{p}(x_{p})=1.$

Since the signature of $x_{\infty}\in\Omega$ is fixed, we have that

$c_{\infty}(x_{\infty})=(-1)^{s(s-1)/2}=:c_{\infty}(\Omega),$

where $s$ is the negative index of the signature of $x$. Hence, we can see
that

$\delta(x)\neq 0\iff c_{2}(x)c_{q}(x)=c_{\infty}(\Omega).$

Recalling that the Tamagawa measure $\mu_{m}$ can be written as

$\mu_{m}=\prod_{\begin{subarray}{c}p\\\ \text{inc.
}\infty\end{subarray}}\mu_{m,p},$

we get the following:

######  Corollary 7.3.

Let $q$ be an odd prime. Then as $q\rightarrow\infty$, we have that

$N(q,\Omega)=(1+o_{\Omega}(1))2q^{\frac{n-1}{2}}\mu_{\infty}(\Omega)\cdot\left(\prod_{p\neq
2,q}\mu_{q,p}(V_{q}(\mathbb{Z}_{p}))\right)\cdot\left(\sum_{\begin{subarray}{c}\mathcal{G}_{2},\mathcal{G}_{q}\\\
c_{2}(\mathcal{G}_{2})c_{q}(\mathcal{G}_{q})=c_{\infty}(\Omega)\end{subarray}}\mu_{q,2}(\mathcal{G}_{2}(\mathbb{Z}_{2}))\mu_{q,q}(\mathcal{G}_{q}(\mathbb{Z}_{q}))\right)$

where for some $\textrm{SL}_{n}(\mathbb{Z}_{p})$-orbit $\mathcal{G}_{p}\subset
V_{q}(\mathbb{Z}_{p})$, we have

$\mu_{p,q}(\mathcal{G}_{p}(\mathbb{Z}_{p}))=\lim_{k\rightarrow\infty}\frac{\\#\mathcal{G}_{p}(\mathbb{Z}/p^{k}\mathbb{Z})}{p^{k(\frac{n(n+1)}{2}-1)}}.$

The sum over $\mathcal{G}_{2}$ (resp. $\mathcal{G}_{q}$) is over the orbits of
$\textrm{SL}_{n}(\mathbb{Z}_{2})$ (resp. $\textrm{SL}_{n}(\mathbb{Z}_{q})$)
inside of $V_{q}(\mathbb{Z}_{2})$ (resp. $V_{q}(\mathbb{Z}_{q})$).

######  Proposition 7.4.

For $n\geq 3$ odd, as the prime $q$ goes to $\infty$, we have that

$\left(\prod_{p\neq
2,q}\mu_{q,p}(V_{q}(\mathbb{Z}_{p}))\right)\cdot\left(\sum_{\begin{subarray}{c}\mathcal{G}_{2},\mathcal{G}_{q}\\\
c_{2}(\mathcal{G}_{2})c_{q}(\mathcal{G}_{q})=c_{\infty}(\Omega)\end{subarray}}\mu_{q,2}(\mathcal{G}_{2}(\mathbb{Z}_{2}))\mu_{q,q}(\mathcal{G}_{q}(\mathbb{Z}_{q}))\right)=\frac{1}{2}\prod_{\begin{subarray}{c}3\leq
j\leq n\\\ j\text{ odd}\end{subarray}}\zeta(j)^{-1}(1+O(q^{-1})).$

For $n\geq 4$ even, as the prime $q$ goes to $\infty$, we have that

$\prod_{p\neq
2,q}\mu_{q,p}(V_{q}(\mathbb{Z}_{p}))\cdot\left(\sum_{\begin{subarray}{c}\mathcal{G}_{2},\mathcal{G}_{q}\\\
c_{2}(\mathcal{G}_{2})c_{q}(\mathcal{G}_{q})=c_{\infty}(\Omega)\end{subarray}}\mu_{q,2}(\mathcal{G}_{2}(\mathbb{Z}_{2}))\mu_{q,q}(\mathcal{G}_{q}(\mathbb{Z}_{q}))\right)=\\\
\frac{1}{2}\prod_{\begin{subarray}{c}3\leq j\leq n\\\ j\text{
odd}\end{subarray}}\zeta(j)^{-1}L(n/2,\genfrac{(}{)}{}{}{(-1)^{n/2}q}{\cdot})\cdot(1+O(q^{-1}))\\\
\times\zeta(n)^{-1}\begin{cases}1,&q\not\equiv(-1)^{n/2}\bmod 4,\\\
1+\frac{2^{1-n}}{1-2^{-n/2}},&q\equiv(-1)^{n/2}\bmod 4,q\equiv\pm 1\bmod 8,\\\
1-\frac{2^{1-n}}{1-2^{-n/2}},&q\equiv(-1)^{n/2}\bmod 4,q\equiv\pm 3\bmod
8.\end{cases}$

###### Proof.

First, we remark that for any finite place $p$, and any
$\textrm{SL}_{n}(\mathbb{Z}_{p})$-orbit $\mathcal{G}$ of matrices inside
$V_{q}(\mathbb{Z}_{p})$, we have that

$\mu_{p,q}(\mathcal{G}(\mathbb{Z}_{p}))=\lim_{k\rightarrow\infty}\frac{\\#\mathcal{G}(\mathbb{Z}/p^{k}\mathbb{Z})}{p^{k(\frac{n(n+1)}{2}-1)}},$

where $\\#\mathcal{G}(\mathbb{Z}/p^{k}\mathbb{Z})$ are the number of matrices
in $V_{q}(\mathbb{Z}/p^{k}\mathbb{Z})$ inside the orbit $\mathcal{G}$.

We begin our analysis with the case of $n$ odd. First, we look at the standard
primes: $p\neq 2,q.$ When dealing with these primes, the action of
$\textrm{SL}_{n}(\mathbb{Z}/p^{k}\mathbb{Z})$ on
$V_{q}(\mathbb{Z}/p^{k}\mathbb{Z})$ given by $S\cdot g=g^{T}Sg$ for
$g\in\textrm{SL}_{n}(\mathbb{Z}/p^{k}\mathbb{Z})$ and $S\in
V_{q}(\mathbb{Z}/p^{k}\mathbb{Z})$ is transitive. Thus, we get that

$\\#V_{q}(\mathbb{Z}/p^{k}\mathbb{Z})=\frac{\\#\textrm{SL}_{n}(\mathbb{Z}/p^{k}\mathbb{Z})}{\\#\\{g\in\textrm{SL}_{n}(\mathbb{Z}/p^{k}\mathbb{Z}):g^{T}Sg=S\\}}.$

For the numerator, it is not hard to see that

$\\#\textrm{SL}_{n}(\mathbb{Z}/p^{k}\mathbb{Z})=p^{k(n^{2}-1)}\prod_{j=2}^{n}(1-p^{-j}).$

On the other hand, for $k$ sufficiently large,

$\\#\\{X\in\textrm{GL}_{n}(\mathbb{Z}/p^{k}\mathbb{Z}):X^{T}SX=S\\}=2\alpha_{p}(S)p^{k\frac{n(n-1)}{2}}.$

This in turn gives us that since $n$ is odd and $p\neq 2,q$,

$\\#\\{g\in\textrm{SL}_{n}(\mathbb{Z}/p^{k}\mathbb{Z}):g^{T}Sg=S\\}=\alpha_{p}(S)p^{k\frac{n(n-1)}{2}}.$

Thus, we have that

$\lim_{k\rightarrow\infty}\frac{\\#V_{q}(\mathbb{Z}/p^{k}\mathbb{Z})}{p^{k(\frac{n(n+1)}{2}-1)}}=\alpha_{p}^{\textrm{std}}(S)^{-1}\prod_{j=2}^{n}(1-p^{-j})=\prod_{\begin{subarray}{c}3\leq
j\leq n\\\ j\text{ odd}\end{subarray}}(1-p^{-j}).$

Next, we handle the case when $n\geq 3$ is odd and $p=q.$ In this case, the
action of $\textrm{SL}_{n}(\mathbb{Z}/q^{k}\mathbb{Z})$ on
$V_{q}(\mathbb{Z}/q^{k}\mathbb{Z})$ is no longer transitive and we have two
disjoint orbits. Let us call the two representatives $S_{1}^{q}$ and
$S_{2}^{q}$. Like before, for $k$ sufficiently large, we can express the
stabilizers in terms of the Siegel masses:

$\\#\\{X\in\textrm{GL}_{n}(\mathbb{Z}/q^{k}\mathbb{Z}):X^{T}S_{i}^{q}X=S_{i}^{q}\\}=2\alpha_{q}(S_{i})q^{k\frac{n(n-1)}{2}}.$

We note that this indeed implies

$\\#\\{g\in\textrm{SL}_{n}(\mathbb{Z}/q^{k}\mathbb{Z}):g^{T}S_{i}^{q}g=S_{i}^{q}\\}=\frac{2\alpha_{q}(S_{i}^{q})q^{k\frac{n(n-1)}{2}}}{\\#\\{x\bmod
q^{k}:x^{2}q\equiv q\bmod
q^{k}\\}}=\alpha_{q}(S_{i}^{q})q^{\frac{kn(n-1)}{2}-1}.$

Hence, for any of the orbits $\mathcal{G}_{q}$:

$\lim_{k\rightarrow\infty}\frac{\\#\mathcal{G}_{q}(\mathbb{Z}/q^{k}\mathbb{Z})}{q^{k(\frac{n(n+1)}{2}-1)}}=q\prod_{j=2}^{n}(1-q^{-j})\alpha_{q}(\mathcal{G}_{q})^{-1}.$

Applying Lemma 6.2, we can see that

$\lim_{k\rightarrow\infty}\frac{\\#\mathcal{G}_{q}(\mathbb{Z}/q^{k}\mathbb{Z})}{q^{k(\frac{n(n+1)}{2}-1)}}=\frac{1}{2}\prod_{\begin{subarray}{c}3\leq
j\leq n\\\ j\text{ odd}\end{subarray}}(1-q^{-j})(1+O(q^{-2})).$

Finally, we look at the case $p=2$. We have two orbits here generated by
$S_{1}^{2}$ and $S_{2}^{2}$, where $\mathcal{O}_{S_{1}}\equiv\pm 1\bmod 8$ and
$\mathcal{O}_{S_{2}}\equiv\pm 3\bmod 8$. We note that both $S_{1}^{2}$ and
$S_{2}^{2}$ will appear in the final sum; in particular, for each of the
$S_{i}^{2}$ there exists exactly one $\textrm{SL}_{n}(\mathbb{Z}_{q})$-orbit,
represented by $S_{j}^{q}$, such that

$c_{2}(S_{i}^{2})c_{q}(S_{j}^{q})=c_{\infty}(\Omega).$

Hence, we get that

$\prod_{p\neq
2,q}\mu_{q,p}(V_{q}(\mathbb{Z}_{p}))\cdot\sum_{\begin{subarray}{c}\mathcal{G}_{2},\mathcal{G}_{q}\\\
c_{2}(\mathcal{G}_{2})c_{q}(\mathcal{G}_{q})=c_{\infty}(\Omega)\end{subarray}}\mu_{q,2}(\mathcal{G}_{2}(\mathbb{Z}_{2}))\mu_{q,q}(\mathcal{G}_{q}(\mathbb{Z}_{q}))=\frac{1}{2}\prod_{\begin{subarray}{c}3\leq
j\leq n\\\ j\text{ odd}\end{subarray}}\prod_{p\neq
2}(1-p^{-j})(1+O(q^{-2}))\cdot\sum_{\mathcal{G}_{2}}\mu_{q,2}(\mathcal{G}_{2}(\mathbb{Z}_{2})),$

where we are summing over the disjoint orbits inside of
$V_{q}(\mathbb{Z}_{2}).$

In order to conclude the odd case of the Proposition, we are left with
evaluating

$\\#V_{q}(\mathbb{Z}/2^{k}\mathbb{Z})=\frac{\\#\textrm{SL}_{n}(\mathbb{Z}/2^{k}\mathbb{Z})}{\\#\\{g\in\textrm{SL}_{n}(\mathbb{Z}/2^{k}\mathbb{Z}):g^{T}S_{1}^{2}g=S_{1}^{2}\\}}+\frac{\\#\textrm{SL}_{n}(\mathbb{Z}/2^{k}\mathbb{Z})}{\\#\\{g\in\textrm{SL}_{n}(\mathbb{Z}/2^{k}\mathbb{Z}):g^{T}S_{2}^{2}g=S_{2}^{2}\\}}.$

We note again that

$\\#\\{X\in\textrm{GL}_{n}(\mathbb{Z}/2^{k}\mathbb{Z}):X^{T}S_{i}^{2}X=S_{i}^{2}\\}=2\alpha_{2}(S_{i}^{2})2^{\frac{kn(n-1)}{2}}.$

For $n$ odd and $k$ sufficiently large, we will have that

$\\#\\{g\in\textrm{SL}_{n}(\mathbb{Z}/2^{k}\mathbb{Z}):g^{T}S_{i}^{2}g=S_{i}^{2}\\}=\frac{1}{4}\\#\\{X\in\textrm{GL}_{n}(\mathbb{Z}/2^{k}\mathbb{Z}):g^{T}S_{i}^{2}g=S_{i}^{2}\\}=\frac{\alpha_{2}(S_{i}^{2})}{2}\cdot
2^{\frac{kn(n-1)}{2}},$

where we have used that the equation $x^{2}\equiv 1\bmod 2^{k}$ has $4$
solutions for $k\geq 3.$

Summing these together, we obtain

$\displaystyle\sum_{\mathcal{G}_{2}}\mu_{q,2}(\mathcal{G}_{2}(\mathbb{Z}_{2}))=\lim_{k\rightarrow\infty}\frac{\\#V_{q}(\mathbb{Z}/2^{k}\mathbb{Z})}{2^{k(\frac{n(n+1)}{2}-1)}}$
$\displaystyle=\prod_{j=2}^{n}(1-2^{-j})(2\alpha_{2}(S_{1}^{2})^{-1}+2\alpha_{2}(S_{2}^{2})^{-1})$
$\displaystyle=\prod_{\begin{subarray}{c}3\leq j\leq n\\\ j\text{
odd}\end{subarray}}(1-2^{-j})\cdot(1-2^{1-n})\cdot\left(\frac{1}{2}(1-2^{\frac{1-n}{2}})^{-1}+\frac{1}{2}(1+2^{\frac{1-n}{2}})^{-1}\right).$

Above, we applied Lemma 6.7. We evaluate this expression to derive:

$\lim_{k\rightarrow\infty}\frac{\\#V_{q}(\mathbb{Z}/2^{k}\mathbb{Z})}{2^{k(\frac{n(n+1)}{2}-1)}}=\prod_{\begin{subarray}{c}3\leq
j\leq n\\\ j\text{ odd}\end{subarray}}(1-2^{-j}).$

Combining all of the factors together, we have the odd case of Proposition
7.4.

We now move on to the even case. The strategy will be the same as before, with
the details being slightly more cumbersome due to the complexities of the
possible Type II Jordan blocks showing up at the prime $2$. We start by
handling the standard primes: $p\neq 2,q.$ Again, we consider the action of
$\textrm{SL}_{n}(\mathbb{Z}/p^{k}\mathbb{Z})$ on
$V_{q}(\mathbb{Z}/p^{k}\mathbb{Z})$ given by $S\cdot g=g^{T}Sg$. Like before,
the action is transitive and since $S$ is diagonalizable ($p$ is a standard
prime), we can assume that $S$ is diagonal. We have

$\\#V_{q}(\mathbb{Z}/p^{k}\mathbb{Z})=\frac{\\#\textrm{SL}_{n}(\mathbb{Z}/p^{k}\mathbb{Z})}{\\#\\{g\in\textrm{SL}_{n}(\mathbb{Z}/p^{k}\mathbb{Z}):g^{T}Sg=S\\}}=\frac{2\cdot\\#\textrm{SL}_{n}(\mathbb{Z}/p^{k}\mathbb{Z})}{\\#\\{g\in\textrm{GL}_{n}(\mathbb{Z}/p^{k}\mathbb{Z}):g^{T}Sg=S\\}},$

where the second equality comes from the fact that since $S$ is diagonal, for
any element of $\textrm{GL}_{n}(\mathbb{Z}/p^{k}\mathbb{Z})$ such that
$g^{T}Sg=S$ and $\det(g)=-1$, we have that $\text{diag}(-1,1,...,1)\cdot
g\in\textrm{SL}_{n}(\mathbb{Z}/p^{k}\mathbb{Z})$ and
$(\text{diag}(-1,1,...,1)\cdot g)^{T}S(\text{diag}(-1,1,...,1)\cdot g)=S$.
Since

$\\#\\{X\in\textrm{GL}_{n}(\mathbb{Z}/p^{k}\mathbb{Z}):X^{T}SX=S\\}=2\alpha_{p}(S)p^{k\frac{n(n-1)}{2}}=2p^{k\frac{n(n-1)}{2}}\prod_{i=1}^{n/2-1}(1-p^{-2k})\cdot(1-p^{-n/2}\genfrac{(}{)}{}{}{(-1)^{n/2}q}{p}),$

we obtain

(14)
$\lim_{k\rightarrow\infty}\frac{\\#V_{q}(\mathbb{Z}/p^{k}\mathbb{Z})}{p^{k(\frac{n(n+1)}{2}-1)}}=\prod_{\begin{subarray}{c}3\leq
j\leq n\\\ j\text{
odd}\end{subarray}}(1-p^{-j})\cdot(1-p^{-n})\left(1-p^{-n/2}\genfrac{(}{)}{}{}{(-1)^{n/2}q}{p}\right)^{-1}.$

Next, we consider the case $p=q$ with $n$ even. There are now two orbits of
the action of $\textrm{SL}_{n}(\mathbb{Z}/q^{k}\mathbb{Z})$ on
$V_{q}(\mathbb{Z}/q^{k}\mathbb{Z})$ and again we can take our two
representatives to be diagonal matrices $S_{1}^{q}$ and $S_{2}^{q}$. We
consider each term individually: For $k$ sufficiently large

$\\#\\{X\in\textrm{GL}_{n}(\mathbb{Z}/q^{k}\mathbb{Z}):X^{T}S_{i}^{q}X=S_{i}^{q}\\}=2\alpha_{q}(S_{i}^{q})q^{k\frac{n(n-1)}{2}},$

which implies

$\\#\\{g\in\textrm{SL}_{n}(\mathbb{Z}/q^{k}\mathbb{Z}):g^{T}S_{i}^{q}g=S_{i}^{q}\\}=\frac{2\alpha_{q}(S_{i}^{q})q^{k\frac{n(n-1)}{2}}}{\\#\\{x\bmod
q^{k}:x^{2}q\equiv q\bmod
q^{k}\\}}=\alpha_{q}(S_{i}^{q})q^{\frac{kn(n-1)}{2}-1}.$

Thus, we have that

$\lim_{k\rightarrow\infty}\frac{\\#\mathcal{G}_{q}(\mathbb{Z}/q^{k}\mathbb{Z})}{q^{k(\frac{n(n+1)}{2}-1)}}=q\prod_{j=2}^{n}(1-q^{-j})\alpha_{q}(\mathcal{G}_{q})^{-1},$

for $\mathcal{G}_{q}$ an $\textrm{SL}_{n}(\mathbb{Z}_{q})$-orbit inside of
$V_{q}(\mathbb{Z}_{q}).$ From Lemma 6.2, we know that

$\alpha_{q}(\mathcal{G}_{q})^{-1}=\frac{1}{2q}\cdot\prod_{j=1}^{n/2-1}(1-q^{-2j})^{-1}\cdot\left(1+O(q^{-2})\right),$

concluding that

(15)
$\lim_{k\rightarrow\infty}\frac{\\#\mathcal{G}_{q}(\mathbb{Z}/q^{k}\mathbb{Z})}{q^{k(\frac{n(n+1)}{2}-1)}}=\frac{1}{2}\prod_{\begin{subarray}{c}3\leq
j\leq n\\\ j\text{
odd}\end{subarray}}(1-q^{-j})\cdot(1-q^{-n})\cdot(1+O(q^{-2})).$

We are left with the trickiest case to consider: $n$ is even and $p=2$. Again,
for each $\textrm{SL}_{n}(\mathbb{Z}_{2})$-orbit there is exactly one
$\textrm{SL}_{n}(\mathbb{Z}_{q})$-orbit, such that the product of the Hasse-
Minkowski invariants is $c_{\infty}(\Omega)$. So, it suffices to calculate

$\sum_{\mathcal{G}_{2}}\mu_{q,2}(\mathcal{G}_{2}(\mathbb{Z}_{2}))=\lim_{k\rightarrow\infty}\frac{\\#V_{q}(\mathbb{Z}/2^{k}\mathbb{Z})}{2^{k(\frac{n(n+1)}{2}-1)}}.$

To see this, we recall that

$\sum\limits_{\mathcal{G}_{2}}\\#\mathcal{G}_{2}(\mathbb{Z}/2^{k}\mathbb{Z})=\\#V_{q}(\mathbb{Z}/2^{k}\mathbb{Z})=\sum_{S_{i}}\frac{\\#\textrm{SL}_{n}(\mathbb{Z}/2^{k}\mathbb{Z})}{\\#\\{X\in\textrm{SL}_{n}(\mathbb{Z}/2^{k}\mathbb{Z}):X^{T}S_{i}X=S_{i}\\}}.$

As we explained earlier

$\\#\\{X\in\textrm{GL}_{n}(\mathbb{Z}/2^{k}\mathbb{Z}):X^{T}S_{i}X=S_{i}\\}=4\\#\\{X\in\textrm{SL}_{n}(\mathbb{Z}/2^{k}\mathbb{Z}):X^{T}S_{i}X=S_{i}\\}.$

Now, if $q\not\equiv(-1)^{n/2}\bmod 4$, then there are again two orbits – both
have $f_{0}$ being Type I, and either we have $\mathcal{O}_{S_{1}}\equiv
0\bmod 8,\mathcal{O}_{S_{2}}\equiv 4\bmod 8$ or $\mathcal{O}_{S_{1}}\equiv
2\bmod 8$ and $\mathcal{O}_{S_{2}}\equiv-2\bmod 8$. Thus, we have that

$\lim_{k\rightarrow\infty}\frac{\\#V_{q}(\mathbb{Z}/2^{k}\mathbb{Z})}{2^{k(\frac{n(n+1)}{2}-1)}}=\prod_{j=2}^{n}(1-2^{-j})\left(2\alpha_{2}(S_{1})^{-1}+2\alpha_{2}(S_{2})^{-1}\right).$

Applying Lemma 6.7, we see that if $\mathcal{O}_{S_{1}}\equiv 0\bmod 8$ and
$\mathcal{O}_{S_{2}}\equiv 4\bmod 8$,

$\alpha_{2}(S_{1})^{-1}+\alpha_{2}(S_{2})^{-1}=\frac{1}{4}\prod_{i=1}^{\frac{n-4}{2}}(1-2^{-2i})^{-1}\left((1-2^{1-n/2})^{-1}+(1+2^{1-n/2})^{-1}\right)=\frac{1}{2}\prod_{i=1}^{\frac{n-2}{2}}(1-2^{-2i})^{-1}.$

Similarly, if $\mathcal{O}_{S_{1}}\equiv 2\bmod 8$ and
$\mathcal{O}_{S_{2}}\equiv-2\bmod 8$, we have

$\alpha_{2}(S_{1})^{-1}+\alpha_{2}(S_{2})^{-1}=\frac{1}{4}\prod_{i=1}^{\frac{n-4}{2}}(1-2^{-2i})^{-1}\left(2\cdot(1-2^{2-n})^{-1}\right)=\frac{1}{2}\prod_{i=1}^{\frac{n-2}{2}}(1-2^{-2i})^{-1}.$

Thus, in the case where $q\not\equiv(-1)^{n/2}\bmod 4$, we have that

(16)
$\lim_{k\rightarrow\infty}\frac{\\#V_{q}(\mathbb{Z}/2^{k}\mathbb{Z})}{2^{k(\frac{n(n+1)}{2}-1)}}=\prod_{\begin{subarray}{c}3\leq
j\leq n\\\ j\text{ odd}\end{subarray}}(1-2^{-j})\cdot(1-2^{-n}).$

If $q\equiv(-1)^{n/2}\bmod 4$ and $q\equiv\pm 1\bmod 8$, there are three
orbits generated by $S_{1},S_{2},S_{3}$ – two of them $S_{1},S_{2}$ are such
that $f_{0}$ is Type I and $\mathcal{O}_{S_{1}}-\mathcal{O}_{S_{2}}\equiv
4\bmod 8$ and the final one $S_{3}$ is such that $f_{0}$ is Type II and
$\mathcal{O}_{S_{3}}\equiv 0\bmod 8.$ We know that

$\lim_{k\rightarrow\infty}\frac{\\#V_{q}(\mathbb{Z}/2^{k}\mathbb{Z})}{2^{k(\frac{n(n+1)}{2}-1)}}=\prod_{j=2}^{n}(1-2^{-j})\left(2\alpha_{2}(S_{1})^{-1}+2\alpha_{2}(S_{2})^{-1}+2\alpha_{2}(S_{3})^{-1}\right),$

and we can compute

$\displaystyle\alpha_{2}(S_{1})^{-1}+\alpha_{2}(S_{2})^{-1}+\alpha_{2}(S_{3})^{-1}$
$\displaystyle=\frac{1}{2}\prod_{i=1}^{\frac{n-2}{2}}(1-2^{-2i})^{-1}+2^{-n}\prod_{i=1}^{\frac{n-2}{2}}(1-2^{-2i})^{-1}\cdot(1-2^{-n/2})^{-1}$
$\displaystyle=\prod_{i=1}^{\frac{n-2}{2}}(1-2^{-2i})^{-1}\left(\frac{1}{2}+\frac{1}{2^{n}(1-2^{-n/2})}\right).$

Hence we obtain

(17)
$\lim_{k\rightarrow\infty}\frac{\\#V_{q}(\mathbb{Z}/2^{k}\mathbb{Z})}{2^{k(\frac{n(n+1)}{2}-1)}}=\prod_{\begin{subarray}{c}3\leq
j\leq n\\\ j\text{
odd}\end{subarray}}(1-2^{-j})\cdot(1-2^{-n})\cdot\left(1+\frac{2^{1-n}}{1-2^{-n/2}}\right).$

Our final case is when $q\equiv(-1)^{n/2}\bmod 4$ and $q\equiv\pm 3\bmod 8.$
We have a similar situation here to the case before, except our three orbits
are generated by $S_{1},S_{2},S_{3}$ – where $S_{1},S_{2}$ have $f_{0}$ Type I
and $\mathcal{O}_{S_{1}}-\mathcal{O}_{S_{2}}\equiv 4\bmod 8$, and $S_{3}$ has
$f_{0}$ Type II and $\mathcal{O}_{S_{3}}\equiv 4\bmod 8.$ Following similar
computations as above, we get that

(18)
$\lim_{k\rightarrow\infty}\frac{\\#V_{q}(\mathbb{Z}/2^{k}\mathbb{Z})}{2^{k(\frac{n(n+1)}{2}-1)}}=\prod_{\begin{subarray}{c}3\leq
j\leq n\\\ j\text{
odd}\end{subarray}}(1-2^{-j})\cdot(1-2^{-n})\cdot\left(1+\frac{2^{1-n}}{1+2^{-n/2}}\right).$

Combining (14), (15), (16), (17), and (18) together, we get the desired
result. ∎

Let us sum up the $N(p,\Omega)$ for $n\geq 3$ odd: applying Corollary 7.3 and
Proposition 7.4, we have that as $T\rightarrow\infty$

$\displaystyle\pi(T\Omega)$ $\displaystyle=\sum_{p\leq T^{n}}N(p,\Omega)$
$\displaystyle=2\mu_{\infty}(\Omega)(1+o_{\Omega}(1))\sum_{p\leq
T^{n}}p^{\frac{n-1}{2}}\cdot\frac{1}{2}\prod_{\begin{subarray}{c}3\leq j\leq
n\\\ j\text{ odd}\end{subarray}}\zeta(j)^{-1}(1+O(p^{-1}))$
$\displaystyle=\mu_{\infty}(\Omega)\prod_{\begin{subarray}{c}3\leq j\leq n\\\
j\text{ odd}\end{subarray}}\zeta(j)^{-1}(1+o_{\Omega}(1))\sum_{p\leq
T^{n}}p^{\frac{n-1}{2}}$
$\displaystyle=\mu_{\infty}(\Omega)\prod_{\begin{subarray}{c}3\leq j\leq n\\\
j\text{
odd}\end{subarray}}\zeta(j)^{-1}\frac{2T^{\frac{n(n+1)}{2}}}{(n+1)\log(T^{n})}(1+o_{\Omega}(1)).$

We remark that Lemma 2.5 tells us that

$\int_{T\Omega}\frac{1}{\log^{+}(\det(x))}dx=\mu_{\infty}(\Omega)\cdot\frac{T^{\dim(V)}}{\dim(V)\log(T)}(1+o(1)).$

Since $\dim(V)=\frac{n(n+1)}{2},$ this gives us the following result.

######  Lemma 7.5.

For $n\geq 3$ odd, and $\Omega$ a compact subset in the orbit
$\mathcal{O}_{p,q}$ with $p,q\neq 0$, we have that

$\pi(T\Omega)=\psi(p,q)\prod_{\begin{subarray}{c}3\leq j\leq n\\\ j\text{
odd}\end{subarray}}(1+o_{\Omega}(1))\int_{T\Omega}\frac{1}{\log^{+}(\det(x))}dx,$

where

$\psi(p,q)=\begin{cases}1,&q\text{ even},\\\ 0,&\text{otherwise.}\end{cases}$

Finally, we end this section by summing up the $N(p,\Omega)$ for $n\geq 4$
even. We utilize that while Proposition 7.4 has many cases, they are
completely determined by the value of $p$ mod 8. So, for $T$ sufficiently
large, we can write:

$\displaystyle\pi(T\Omega)$ $\displaystyle=\sum_{\begin{subarray}{c}p\leq
T^{n}\\\ p\not\equiv(-1)^{n/2}\bmod
4\end{subarray}}N(p,\Omega)+\sum_{\begin{subarray}{c}p\leq T^{n}\\\
p\equiv(-1)^{n/2}\bmod 4\\\ p\equiv\pm 1\bmod
8\end{subarray}}N(p,\Omega)+\sum_{\begin{subarray}{c}p\leq T^{n}\\\
p\equiv(-1)^{n/2}\bmod 4\\\ p\equiv\pm 3\bmod 8\end{subarray}}N(p,\Omega)$
$\displaystyle=\frac{1}{2}\psi(p,q)\mu_{\infty}(\Omega)\zeta(n)^{-1}\prod_{\begin{subarray}{c}3\leq
j\leq n\\\ j\text{ odd}\end{subarray}}\zeta(j)^{-1}(1+o_{\Omega}(1))$
$\displaystyle\hskip 28.45274pt\times\Bigg{(}\sum_{\begin{subarray}{c}p\leq
T^{n}\\\ p\not\equiv(-1)^{n/2}\bmod
4\end{subarray}}L(n/2,\chi_{p})p^{\frac{n-1}{2}}+\sum_{\begin{subarray}{c}p\leq
T^{n}\\\ p\equiv(-1)^{n/2}\bmod 4\\\ p\equiv\pm 1\bmod
8\end{subarray}}L(n/2,\chi_{p})p^{\frac{n-1}{2}}\cdot(1+\frac{2^{1-n}}{1-2^{-n/2}})$
$\displaystyle\hskip 113.81102pt+\sum_{\begin{subarray}{c}p\leq T^{n}\\\
p\equiv(-1)^{n/2}\bmod 4\\\ p\equiv\pm 3\bmod
8\end{subarray}}L(n/2,\chi_{p})p^{\frac{n-1}{2}}(1+\frac{2^{1-n}}{1+2^{-n/2}})\Bigg{)},$

where $\chi_{p}=\genfrac{(}{)}{}{}{(-1)^{n/2}p}{\cdot}.$ To evaluate the above
sums, it suffices to determine an asymptotic for the following:

######  Lemma 7.6.

For $n\geq 4$, we have that for $(a,8)=1$ as $T\rightarrow\infty$,

$\sum_{\begin{subarray}{c}p\leq T^{n}\\\ p\equiv a\bmod
8\end{subarray}}L(n/2,\chi_{p})p^{\frac{n-1}{2}}=\frac{\zeta(n)}{4}\cdot\left(1-2^{-n}\right)\cdot\left(\frac{2T^{\frac{n(n+1)}{2}}}{n(n+1)\log(T)}(1+o(1)\right).$

###### Proof.

Since $n\geq 4$, we can expand out the definition of $L(n/2,\chi_{p})$:

$\displaystyle\sum_{\begin{subarray}{c}p\leq T^{n}\\\ p\equiv a\bmod
8\end{subarray}}p^{\frac{n-1}{2}}\sum_{\begin{subarray}{c}m=1\\\
(m,2p)=1\end{subarray}}^{\infty}\genfrac{(}{)}{}{}{(-1)^{n/2}p}{m}\frac{1}{m^{n/2}}$
$\displaystyle=\sum_{\begin{subarray}{c}m=1\\\
(m,2)=1\end{subarray}}^{\infty}\frac{1}{m^{n/2}}\sum_{\begin{subarray}{c}p\leq
T^{n}\\\ p\equiv a\bmod
8\end{subarray}}p^{\frac{n-1}{2}}\genfrac{(}{)}{}{}{(-1)^{n/2}p}{m}.$

We note that since $m$ is odd, the character
$\genfrac{(}{)}{}{}{(-1)^{n/2}p}{m}$ is trivial if and only if $m$ is a
square. In this case, we have the contribution to the sum is:

$\sum_{\begin{subarray}{c}\ell=1\\\
(\ell,2)=1\end{subarray}}^{\infty}\frac{1}{\ell^{n}}\sum_{\begin{subarray}{c}p\leq
T^{n}\\\ p\equiv a\bmod
8\end{subarray}}p^{\frac{n-1}{2}}=\frac{\zeta(n)}{4}\cdot(1-2^{-n})\cdot\left(\frac{2T^{\frac{n(n+1)}{2}}}{n(n+1)\log(T)}+O(T^{\frac{n(n+1)}{2}}\log(T)^{-101})\right).$

On the other hand, if $m$ is not a square and $m\geq\log(T)^{101}$, we have
that since $n\geq 4$,

$\sum_{m\geq\log(T)^{101}}\frac{1}{m^{n/2}}\sum_{\begin{subarray}{c}p\leq
T^{n}\\\ p\equiv a\bmod
8\end{subarray}}\genfrac{(}{)}{}{}{(-1)^{n/2}p}{m}p^{\frac{n-1}{2}}\ll\frac{T^{\frac{n(n+1)}{2}}}{\log(T)}\sum_{m\geq\log(T)^{101}}\frac{1}{m^{n/2}}\ll
T^{\frac{n(n+1)}{2}}\log(T)^{-102}.$

Finally, if $m$ is not a square, the character is nontrivial and we have that

$\sum_{\begin{subarray}{c}p\leq T^{n}\\\ p\equiv a\bmod
8\end{subarray}}\genfrac{(}{)}{}{}{(-1)^{n/2}p}{m}p^{\frac{n-1}{2}}\ll
T^{\frac{n(n+1)}{2}}\log(T)^{-101},$

for $m\leq\log(T)^{101}$. Together, we get the lemma. ∎

Plugging this into $\pi(T\Omega)$, we get:

$\displaystyle\pi(T\Omega)$
$\displaystyle=\psi(p,q)\mu_{\infty}(\Omega)\prod_{\begin{subarray}{c}3\leq
j\leq n\\\ j\text{
odd}\end{subarray}}\zeta(j)^{-1}(1+o_{\Omega}(1))\cdot\frac{2T^{\frac{n(n+1)}{2}}}{n(n+1)\log(T)}$
$\displaystyle\hskip
28.45274pt\times\Bigg{(}\frac{1}{2}(1-2^{-n})+\frac{1}{4}(1-2^{-n})(1+\frac{2^{1-n}}{1-2^{-n/2}})+\frac{1}{4}(1-2^{-n})(1+\frac{2^{1-n}}{1+2^{-n/2}})\Bigg{)}$
$\displaystyle=\psi(p,q)\mu_{\infty}(\Omega)\prod_{\begin{subarray}{c}3\leq
j\leq n\\\ j\text{
odd}\end{subarray}}\zeta(j)^{-1}\cdot\frac{2T^{\frac{n(n+1)}{2}}}{n(n+1)\log(T)}\cdot(1+o_{\Omega}(1)).$

Thus, we get the following lemma.

######  Lemma 7.7.

For $n\geq 3$ even, and $\Omega$ a nice compact subset in the orbit
$\mathcal{O}_{p,q}$ with $p,q\neq 0$, we have that as $T\rightarrow\infty$,

$\pi(T\Omega)=\psi(p,q)\prod_{\begin{subarray}{c}3\leq j\leq n\\\ j\text{
odd}\end{subarray}}(1+o_{\Omega}(1))\int_{T\Omega}\frac{1}{\log^{+}(\det(x))}dx.$

###  7.2. The positive-definite orbit

For positive definite symmetric matrices, we can no longer apply Oh’s result,
since the stabilizer $\textrm{SO}_{n}(\mathbb{R})$ is a compact group.
Instead, we appeal to an equidistribution result of Einsiedler, Margulis,
Mohammadi, and Venkatesh [13]. We will proceed for the case of positive
definite matrices $\mathcal{O}_{n,0}$ (which produce positive primes), as the
negative definite case is identical.

####  7.2.1. Linnik equidistribution

Let
$\mathcal{M}=\textrm{PGL}_{n}(\mathbb{Z})\backslash\textrm{PGL}_{n}(\mathbb{R})/\textrm{PO}_{n}(\mathbb{R})$
be the space of $\textrm{PGL}_{n}(\mathbb{Z})$-equivalence classes of positive
definite quadratic forms over $\mathbb{R}^{n}$. We equip $\mathcal{M}$ with
the pushforward of the Haar measure on $\textrm{PGL}_{n}(\mathbb{Z})\backslash
P\textrm{GL}_{n}(\mathbb{R})$. We now quote the main equidistribution result
that we require from Einsiedler, Margulis, Mohammadi, and Venkatesh [13]:

######  Theorem 7.8 (Einsiendler, Margulis, Mohammadi, Venkatesh, [13] Theorem
3.1).

Suppose $\\{Q_{i}\\}_{i=1}^{\infty}$ varies through any sequence of pairwise
inequivalent, integral, positive definite quadratic forms. Then the genus of
$Q_{i}$, considered as a subset of
$\textrm{PGL}_{n}(\mathbb{Z})\backslash\textrm{PGL}_{n}(\mathbb{R})/\textrm{PO}_{n}(\mathbb{R})$,
equidistributes as $i\rightarrow\infty.$

Recall that two integral quadratic forms in the same genus have the same
determinant. As a corollary we have the following:

######  Corollary 7.9.

Let
$\mathcal{M}=\textrm{PGL}_{n}(\mathbb{Z})\backslash\textrm{PGL}_{n}(\mathbb{R})/\textrm{PO}_{n}(\mathbb{R})$
be the space of positive definite quadratic forms of determinant $1$ up to
$\mathbb{Z}$-equivalence, equipped with $\mu$, the pushforward of the Haar
measure on
$\textrm{PGL}_{n}(\mathbb{Z})\backslash\textrm{PGL}_{n}(\mathbb{R})$
normalized so that $\mu(\mathcal{M})=1$. Let $Q_{m}\subset\mathcal{M}$ denote
the family of integral quadratic forms of determinant equal to $m$. If
$\mu_{m}$ denotes the normalized Dirac measure over the family $Q_{m}$, then
as $m\rightarrow\infty$:

$\mu_{m}\xrightarrow{\text{$w^{*}$}}\mu.$

###### Proof.

We note that for a fixed value of $m$, since $Q_{m}$ contains all integral
quadratic forms of determinant $m$, we can write it as

$Q_{m}=\cup_{\mathcal{G}_{i}}Q_{m,\mathcal{G}_{i}},$

where $\mathcal{G}_{i}$ runs through the set of genera with determinant $m$
and $Q_{m,\mathcal{G}_{i}}$ is a set containing all of the inequivalent
integral forms inside of $\mathcal{G}_{i}$. Applying Theorem 7.8 to
$Q_{m,\mathcal{G}_{i}}$, we have that these genera will equidistribute as
$m\rightarrow\infty$. In otherwords, if $\mu_{m,\mathcal{G}_{i}}$ denotes the
Dirac measure on $Q_{m,\mathcal{G}_{i}}$, then we have that as
$m\rightarrow\infty$,

$\frac{\mu_{m,\mathcal{G}_{i}}}{\\#Q_{m,\mathcal{G}_{i}}}\xrightarrow{w^{*}}\mu.$

This then implies that $\frac{\mu_{m}}{\\#Q_{m}}\xrightarrow{w^{*}}\mu$ as
desired. ∎

We can naturally identify
$\textrm{PGL}_{n}(\mathbb{Z})\backslash\textrm{PGL}_{n}(\mathbb{R})/\textrm{PO}_{n}(\mathbb{R})$
with the space
$\textrm{SL}_{n}(\mathbb{Z})\backslash\textrm{SL}_{n}(\mathbb{R})/\textrm{SO}_{n}(\mathbb{R})$.
Under this identification we have that for any compact
$\Omega_{0}\subset\textrm{SL}_{n}(\mathbb{Z})\backslash\textrm{SL}_{n}(\mathbb{R})/\textrm{SO}_{n}(\mathbb{R})$,

$\lim_{m\rightarrow\infty}\frac{\\#\\{A\in\Omega_{0}:m^{\frac{1}{n}}A\in\textrm{Mat}_{n}(\mathbb{Z})\\}}{h_{n}(m)}\rightarrow\mu(\Omega_{0}),$

where $h_{n}(m)$ is the class number of
$\textrm{SL}_{n}(\mathbb{Z})$-conjugacy orbits of integral positive definite
symmetric matrices of determinant $m$.

Next, we note that for any compact
$\Omega\subset\textrm{SL}_{n}(\mathbb{R})/\textrm{SO}_{n}(\mathbb{R})$, we can
decompose

$\Omega=\cup_{i=1}^{k}\Omega_{i},$

where $\Omega_{i}$ are distinct and each contained in different copies of the
fundamental domain for the action of $\textrm{SL}_{n}(\mathbb{Z})$ on
$\textrm{SL}_{n}(\mathbb{R})/\textrm{SO}_{n}(\mathbb{R}).$ Then if we define a
lift $\overline{\mu}$ of $\mu$ as

$\overline{\mu}(\Omega)=\sum_{i=1}^{k}\mu(\Omega_{i}),$

we have that

(19)
$\lim_{m\rightarrow\infty}\frac{\\#\\{A\in\Omega:m^{1/n}A\in\textrm{Mat}_{n}(\mathbb{Z})\\}}{h_{n}(m)}\rightarrow\overline{\mu}(\Omega).$

We will abuse notation and denote $\overline{\mu}$ as $\mu$ from now on.

####  7.2.2. Computing the class number

Now, we will compute the class number $h_{n}(p)$ as $p$ ranges over primes. We
note that for $n$ odd, this is computed by Kitaoka in [21]. We start at the
same point, with a result of Kitaoka relating the class number to Siegel
masses:

######  Theorem 7.10 ([21], Theorem 2).

Let $n\geq 2$. Then as $m\rightarrow\infty$, we have that

$h_{n}(m)=(1+o(1))\cdot\begin{cases}2\sum_{\mathcal{G}_{i}}M(\mathcal{G}_{i}),&n\text{
odd},\\\ 4\sum_{\mathcal{G}_{i}}M(\mathcal{G}_{i}),&n\text{ even}.\end{cases}$

Here the sum ranges over the genera of positive definite quadratic forms of
determinant $m$.

Using the above, we derive the following estimate for $h_{n}(q)$ for $q$
prime:

######  Proposition 7.11.

Let $n\geq 3$. Then for $n$ odd, as $q\rightarrow\infty$ for $q$ prime, we
have that:

$h_{n}(q)=(1+o(1))\frac{q^{(n-1)/2}}{\pi^{n(n+1)/4}}\prod_{k=1}^{n}\Gamma(k/2)\prod_{k=1}^{\frac{n-1}{2}}\zeta(2k).$

For $n\geq 4$ even, as the prime $q$ goes to $\infty$, we have that

$h_{n}(q)=(1+o(1))\frac{q^{\frac{n-1}{2}}}{\pi^{\frac{n(n+1)}{4}}}\prod_{k=1}^{n}\Gamma(k/2)\prod_{k=1}^{\frac{n-1}{2}}\zeta(2k)L(n/2,\genfrac{(}{)}{}{}{(-1)^{n/2}q}{\cdot}\\\
\times\begin{cases}2,&q\not\equiv(-1)^{n/2}\bmod 4\\\
2+\frac{2^{-n+2}}{1-2^{-n/2}},&q\equiv(-1)^{n/2}\bmod 4,q\equiv\pm 1\bmod 8\\\
2+\frac{2^{-n+2}}{1+2^{-n/2}},&q\equiv(-1)^{n/2}\bmod 4,q\equiv\pm 3\bmod
8.\end{cases}$

######  Remark 7.12.

We note that in the case of $n$ odd, this asymptotic was previously derived by
Kitaoka in [21].

###### Proof.

First, we handle the case that $n$ is odd. Using the Siegel mass formula and
(10), we know that for any genus $\mathcal{G}$, we have that

$M(\mathcal{G})=2\frac{q^{\frac{n+1}{2}}}{\pi^{\frac{n(n+1)}{4}}}\prod_{k=1}^{n}\Gamma(k/2)\prod_{p\neq
2,q}\left(\prod_{k=1}^{\frac{n-1}{2}}(1-p^{-2k})^{-1}\right)\cdot\alpha_{q}(\mathcal{G})^{-1}\alpha_{2}(\mathcal{G})^{-1}.$

Now, Lemma 6.2 tells us that

$\alpha_{q}(\mathcal{G})^{-1}=\frac{1}{2q}\prod_{k=1}^{\frac{n-1}{2}}(1-q^{-2k})^{-1}\cdot(1+O(q^{-1})).$

On the other hand, Lemma 6.7 tells us that

$\alpha_{2}(\mathcal{G})^{-1}=\frac{1}{4}\prod_{k=1}^{\frac{n-3}{2}}(1-2^{-2k})^{-1}\times\begin{cases}(1-2^{\frac{1-n}{2}})^{-1},&\mathcal{O}_{\mathcal{G}}\equiv\pm
1\bmod 8\\\ (1+2^{\frac{1-n}{2}})^{-1},&\mathcal{O}_{\mathcal{G}}\equiv\pm
3\bmod 8.\end{cases}$

We know that there are two $\textrm{SL}_{n}(\mathbb{Z}_{2})$-orbits
$\mathcal{G}_{1}$ and $\mathcal{G}_{2}$ of determinant $q$, one satisfying
$\mathcal{O}_{\mathcal{G}_{1}}\equiv\pm 1\bmod 8$ and the other satisfying
$\mathcal{O}_{\mathcal{G}_{2}}\equiv\pm 3\bmod 8.$ This follows from the fact
that there are two classes over $\mathbb{Z}_{q}$ and two classes over
$\mathbb{Z}_{2}$, and the quadratic reciprocity relation (see Cassels [6],
Chapter 9). Thus, we have that

$\displaystyle h_{n}(q)$
$\displaystyle=(1+o(1))\cdot\frac{q^{(n-1)/2}}{2\pi^{n(n+1)/4}}\prod_{k=1}^{n}\Gamma(k/2)\prod_{k=1}^{\frac{n-1}{2}}\zeta(2k)\cdot(1-2^{1-n})\left(\frac{1}{1-2^{\frac{1-n}{2}}}+\frac{1}{1+2^{\frac{1-n}{2}}}\right)$
$\displaystyle=(1+o(1))\cdot\frac{q^{(n-1)/2}}{\pi^{n(n+1)/4}}\prod_{k=1}^{n}\Gamma(k/2)\prod_{k=1}^{\frac{n-1}{2}}\zeta(2k).$

Next, we handle the more involved case where $n$ is even. First, we consider
when $q\not\equiv(-1)^{n/2}\bmod 4.$ Again applying the Siegel mass formula,
(10), Lemma 6.2, and Lemma 6.7, we have that for $\mathcal{G}$ of Type I:

$M(\mathcal{G})=\frac{q^{\frac{n-1}{2}}}{\pi^{n(n+1)/4}}\prod_{k=1}^{n}\Gamma(k/2)\prod_{k=1}^{\frac{n}{2}-1}\zeta(2k)L(n/2,\genfrac{(}{)}{}{}{(-1)^{n/2}q}{\cdot})\\\
\times\frac{1}{4}(1-2^{2-n})\cdot\begin{cases}(1-2^{1-\frac{n}{2}})^{-1},&\mathcal{O}_{\mathcal{G}}\equiv
0\bmod 8\\\ (1+2^{1-\frac{n}{2}})^{-1},&\mathcal{O}_{\mathcal{G}}\equiv 4\bmod
8\\\ (1-2^{2-n})^{-1},&\mathcal{O}_{\mathcal{G}}\equiv\pm 2\bmod
8.\end{cases}$

Again, we have two $\textrm{SL}_{n}(\mathbb{Z}_{2})$-orbits $\mathcal{G}_{1}$
and $\mathcal{G}_{2}$ that satisfy
$\mathcal{O}_{\mathcal{G}_{1}}-\mathcal{O}_{\mathcal{G}_{2}}\equiv 4\bmod 8$.
Hence we have that either

$h_{n}(q)=(1+o(1))\cdot\frac{q^{(n-1)/2}}{\pi^{n(n+1)/4}}\prod_{k=1}^{n}\Gamma(k/2)\prod_{k=1}^{n/2-1}\zeta(2k)L(n/2,\genfrac{(}{)}{}{}{(-1)^{n/2}q}{\cdot})\\\
\times(1-2^{2-n})\left(\frac{1}{1-2^{1-n/2}}+\frac{1}{1+2^{1-n/2}}\right)$

or

$h_{n}(q)=(1+o(1))\cdot\frac{q^{(n-1)/2}}{\pi^{n(n+1)/4}}\prod_{k=1}^{n}\Gamma(k/2)\prod_{k=1}^{n/2-1}\zeta(2k)L(n/2,\genfrac{(}{)}{}{}{(-1)^{n/2}q}{\cdot})\\\
\times(1-2^{2-n})\left(\frac{1}{1-2^{2-n}}+\frac{1}{1-2^{2-n}}\right).$

In both cases:

$h_{n}(q)=(2+o(1))\cdot\frac{q^{(n-1)/2}}{\pi^{n(n+1)/4}}\prod_{k=1}^{n}\Gamma(k/2)\prod_{k=1}^{n/2-1}\zeta(2k)L(n/2,\genfrac{(}{)}{}{}{(-1)^{n/2}q}{\cdot}).$

Second, we consider when $q\equiv(-1)^{n/2}\bmod 4$ and $q\equiv\pm 1\bmod 8.$
Now we will have three $\textrm{SL}_{n}(\mathbb{Z}_{2})$-orbits –
$\mathcal{G}_{1},\mathcal{G}_{2},\mathcal{G}_{3}$ where $\mathcal{G}_{1,2}$
have $f_{0}$ Type I and
$\mathcal{O}_{\mathcal{G}_{1}}-\mathcal{O}_{\mathcal{G}_{2}}\equiv 4\bmod 8$
and $\mathcal{G}_{3}$ has $f_{0}$ Type II with
$\mathcal{O}_{\mathcal{G}_{3}}\equiv 0\bmod 8.$ Applying the Siegel mass
formula, (10), Lemma 6.2, and Lemma 6.7, we have an extra contribution from
$\mathcal{G}_{3}$:

$M(\mathcal{G}_{3})=\frac{q^{(n-1)/2}}{\pi^{n(n+1)/4}}\prod_{k=1}^{n}\Gamma(k/2)\prod_{k=1}^{\frac{n}{2}-1}\zeta(2k)L(n/2,\genfrac{(}{)}{}{}{(-1)^{n/2}q}{\cdot})2^{-n}(1-2^{-n/2})^{-1}.$

Combining this with the calculations above, we see that

$h_{n}(q)=(1+o(1))\cdot(2+\frac{2^{-n+2}}{1-2^{-n/2}})\cdot\frac{q^{(n-1)/2}}{\pi^{n(n+1)/4}}\prod_{k=1}^{n}\Gamma(k/2)\prod_{k=1}^{n/2-1}\zeta(2k)L(n/2,\genfrac{(}{)}{}{}{(-1)^{n/2}q}{\cdot}).$

Our final case is when $q\equiv(-1)^{n/2}\bmod 4$ and $q\equiv\pm 3\bmod 8.$
We will have three $\textrm{SL}_{n}(\mathbb{Z}_{2})$-orbits –
$\mathcal{G}_{1},\mathcal{G}_{2},\mathcal{G}_{3}$ where $\mathcal{G}_{1,2}$
have $f_{0}$ Type I and
$\mathcal{O}_{\mathcal{G}_{1}}-\mathcal{O}_{\mathcal{G}_{2}}\equiv 4\bmod 8$
and $\mathcal{G}_{3}$ has $f_{0}$ Type II with
$\mathcal{O}_{\mathcal{G}_{3}}\equiv 4\bmod 8.$ Applying the Siegel mass
formula, (10), Lemma 6.2, and Lemma 6.7, we have that the extra contribution
from $\mathcal{G}_{3}$ is given by

$M(\mathcal{G}_{3})=\frac{q^{(n-1)/2}}{\pi^{n(n+1)/4}}\prod_{k=1}^{n}\Gamma(k/2)\prod_{k=1}^{\frac{n}{2}-1}\zeta(2k)L(n/2,\genfrac{(}{)}{}{}{(-1)^{n/2}q}{\cdot})2^{-n}(1+2^{-n/2})^{-1}.$

Hence, combining with the computations above, we get that

$h_{n}(q)=(1+o(1))\cdot(2+\frac{2^{-n+2}}{1+2^{-n/2}})\cdot\frac{q^{(n-1)/2}}{\pi^{n(n+1)/4}}\prod_{k=1}^{n}\Gamma(k/2)\prod_{k=1}^{n/2-1}\zeta(2k)L(n/2,\genfrac{(}{)}{}{}{(-1)^{n/2}p}{\cdot}).$

∎

####  7.2.3. Comparison of the measures

Before completing this section, we need to compare the Haar measure $\mu$ from
(19) to the Bateman-Horn measure $\mu_{\infty}.$ Let us first place our
measures on the same space. Let us notate the following map:

$\psi:\textrm{SL}_{n}(\mathbb{R})/\textrm{SO}_{n}(\mathbb{R})\rightarrow\mathcal{O}_{n,0}\subset\mathbb{R}^{\frac{n(n+1)}{2}}$
$X\mapsto XX^{T}.$

Notice that this map is bijective and hence we can identify these spaces.
Denote by $\mu^{SO_{n}}$ the Riemannian measure on
$\textrm{SO}_{n}(\mathbb{R})$ used by Zhang in [34]. Under this measure, the
volume of $\textrm{SO}_{n}(\mathbb{R})$ is computed in [34] as

$\mu^{SO_{n}}(\textrm{SO}_{n}(\mathbb{R}))=2^{n-1}\pi^{\frac{n(n+1)}{4}}\prod_{k=1}^{n}\Gamma(k/2)^{-1}.$

Denote further by $\mu_{0}^{SO_{n}}$ the normalization of this measure:

$\mu_{0}^{\textrm{SO}_{n}(\mathbb{R})}(\textrm{SO}_{n}(\mathbb{R})):=2^{-n+1}\pi^{\frac{-n(n+1)}{4}}\prod_{k=1}^{n}\Gamma(k/2)\mu^{\textrm{SO}_{n}(\mathbb{R})}.$

We will use $\mu^{\textrm{SO}_{n}}$ and $\mu_{0}^{\textrm{SO}_{n}}$ to relate
the measures $\mu$ and $\mu_{\infty}.$

######  Lemma 7.13.

For any compact set
$\Omega\subset\textrm{SL}_{n}(\mathbb{R})/\textrm{SO}_{n}(\mathbb{R})$, we
have that if $n$ is odd, then

$\mu(\Omega)=\pi^{\frac{n(n+1)}{4}}\prod_{j=2}^{n}\zeta(j)^{-1}\prod_{k=1}^{n}\Gamma(k/2)^{-1}\mu_{\infty}(\Omega).$

If $n$ is even, then

$\mu(\Omega)=\frac{1}{2}\pi^{\frac{n(n+1)}{4}}\prod_{j=2}^{n}\zeta(j)^{-1}\prod_{k=1}^{n}\Gamma(k/2)^{-1}\mu_{\infty}(\Omega).$

###### Proof.

Consider the following map:

$\phi:\mathcal{O}_{n,0}\times\textrm{SO}_{n}(\mathbb{R})\rightarrow\textrm{SL}_{n}(\mathbb{R})$
$(S,Q)\mapsto\sqrt{S}Q.$

Since $S$ is an identification as a positive definite symmetric matrix and
positive definite symmetric matrices have unique square roots, this map is
well-defined. Additionally, the map $\phi$ is bijective and the composition

$\mathcal{O}_{n,0}\times\textrm{SO}_{n}(\mathbb{R})\xrightarrow{\phi}\textrm{SL}_{n}(\mathbb{R})\xrightarrow{X\mapsto
XX^{T}}\mathcal{O}_{n,0}$

sends $(S,Q)\mapsto S.$

Let $\mu_{\infty}^{\textrm{SL}_{n}}$ be the Bateman-Horn measure on
$\textrm{SL}_{n}(\mathbb{R})$ as computed in Lemma 2.5. We remark that this is
a Haar measure on $\textrm{SL}_{n}(\mathbb{R}).$ We also know that
$\phi_{*}(d\mu\times d\mu_{0}^{\textrm{SO}_{n}(\mathbb{R})})$ is a Haar
measure on $\textrm{SL}_{n}(\mathbb{R})$ and the Haar measure on
$\textrm{SL}_{n}(\mathbb{R})$ is unique up to scalar multiple. We wish to
compute the scalar between these two measures; we do so by computing the
volumes of the two measures on a fundamental domain of
$\textrm{SL}_{2}(\mathbb{Z})\backslash\textrm{SL}_{n}(\mathbb{R})$.

Let $\mathcal{F}\subset\textrm{SL}_{n}(\mathbb{R})$ be a fundamental domain
for
$\textrm{SL}_{n}(\mathbb{Z})\backslash\textrm{SL}_{n}(\mathbb{R})/\textrm{SO}_{n}(\mathbb{R})$
and $\mathcal{F}^{\prime}\subset\textrm{SL}_{n}(\mathbb{R})$ be a fundamental
domain for $\textrm{SL}_{n}(\mathbb{Z})\backslash\textrm{SL}_{n}(\mathbb{R})$.
We remark that for almost all $A\in\textrm{SL}_{n}(\mathbb{R})$, we can
compute that

$A^{-1}\textrm{SL}_{n}(\mathbb{Z})A\cap\textrm{SO}_{n}(\mathbb{R})=c_{n}=\begin{cases}1,&n\text{
odd},\\\ 2,&n\text{ even.}\end{cases}$

Hence, we have that

$\int_{\mathcal{F}^{\prime}}\phi_{*}(d\mu\times
d\mu_{0}^{\textrm{SO}_{n}(\mathbb{R})})=c_{n}^{-1}\mu(\mathcal{F})\mu_{0}^{\textrm{SO}_{n}(\mathbb{R})}(\textrm{SO}_{n}(\mathbb{R}))=\begin{cases}1,&n\text{
odd},\\\ 1/2,&n\text{ even.}\end{cases}$

Here we have recalled that $\mu$ was normalized such that
$\mu(\mathcal{F})=1.$ However, Appendix A of [12] computes that

$\mu_{\infty}^{\textrm{SL}_{n}}(\mathcal{F}^{\prime})=\mu_{\infty}^{\textrm{SL}_{n}}(\textrm{SL}_{n}(\mathbb{Z})\backslash\textrm{SL}_{n}(\mathbb{R}))=\prod_{j=2}^{n}\zeta(j),$

and so we get the relation

$\displaystyle\phi_{*}(d\mu\times d\mu_{0}^{\textrm{SO}_{n}(\mathbb{R})})$
$\displaystyle=\frac{1}{c_{n}\prod_{j=2}^{n}\zeta(j)}d\mu_{\infty}^{\textrm{SL}_{n}(\mathbb{R})}.$

On the other hand, by Lemma 2.27 in [34]

$\displaystyle d\mu_{\infty}\wedge d\mu^{\textrm{SO}_{n}(\mathbb{R})}$
$\displaystyle=2^{n-1}d\mu_{\infty}^{\textrm{SL}_{n}(\mathbb{R})}.$

Putting these two together, we achieve that

$\phi_{*}(d\mu\times
d\mu_{0}^{\textrm{SO}_{n}(\mathbb{R})})=\frac{1}{c_{n}2^{n-1}\prod_{j=2}^{n}\zeta(j)}d\mu_{\infty}\wedge
d\mu^{\textrm{SO}_{n}(\mathbb{R})}\\\
=2^{n-1}\pi^{\frac{n(n+1)}{4}}\prod_{k=1}^{n}\Gamma(k/2)^{-1}\frac{1}{c_{n}2^{n-1}\prod_{j=2}^{n}\zeta(j)}d\mu_{\infty}\wedge
d\mu_{0}^{\textrm{SO}_{n}(\mathbb{R})},$

and hence

$d\mu=c_{n}^{-1}\pi^{\frac{n(n+1)}{4}}\prod_{j=2}^{n}\zeta(j)^{-1}\prod_{k=1}^{n}\Gamma(k/2)^{-1}d\mu_{\infty}.$

This completes the proof. ∎

####  7.2.4. Proof of Theorem 1.8

Finally, we can combine all of these results together. First we consider when
$n$ is odd. Taking together (19), Proposition 7.11, and Lemma 7.13, we have
that as $p\rightarrow\infty$,

$N(p,\Omega)=(1+o_{\Omega}(1))\mu_{\infty}(\Omega)\prod_{\begin{subarray}{c}3\leq
j\leq n\\\ j\text{ odd}\end{subarray}}\zeta(j)^{-1}p^{\frac{n-1}{2}}.$

Summing over $p$, we have the following result.

######  Lemma 7.14.

For $n\geq 3$ odd, and $\Omega\subset\mathcal{O}_{n,0}$ a compact subset, we
have that as $T\rightarrow\infty$,

$\pi(T\Omega)=(1+o_{\Omega}(1))\prod_{\begin{subarray}{c}3\leq j\leq n\\\
j\text{
odd}\end{subarray}}\zeta(j)^{-1}\int_{T\Omega}\frac{1}{\log^{+}(\det(x))}dx.$

Next, we deal with the case when $n$ is even. We can see that (19),
Proposition 7.11, and Lemma 7.13 combine to form:

$N(p,\Omega)=(1+o_{\Omega}(1))\mu_{\infty}(\Omega)\prod_{\begin{subarray}{c}3\leq
j\leq n\\\ j\text{
odd}\end{subarray}}\zeta(j)^{-1}\cdot\zeta(n)^{-1}L(n/2,\genfrac{(}{)}{}{}{(-1)^{n/2}p}{\cdot})\\\
\times\begin{cases}1,&p\not\equiv(-1)^{n/2}\bmod 4\\\
1+\frac{2^{-n+1}}{1-2^{-n/2}},&p\equiv(-1)^{n/2}\bmod 4,p\equiv\pm 1\bmod 8\\\
1+\frac{2^{-n+1}}{1+2^{-n/2}},&p\equiv(-1)^{n/2}\bmod 4,p\equiv\pm 3\bmod
8.\end{cases}$

Now we apply Lemma 7.6 to see that

$\displaystyle\pi(T\Omega)$
$\displaystyle=(1+o_{\Omega}(1))\mu_{\infty}(\Omega)\frac{2T^{n(n+1)/2}}{n(n+1)\log(T)}\prod_{\begin{subarray}{c}3\leq
j\leq n\\\ j\text{
odd}\end{subarray}}\zeta(j)^{-1}\left((1-2^{-n})\cdot\left(1+\frac{2^{-n+1}}{4(1-2^{-n/2})}+\frac{2^{-n+1}}{4(1+2^{-n/2})}\right)\right)$
$\displaystyle=(1+o_{\Omega}(1))\mu_{\infty}(\Omega)\frac{2T^{n(n+1)/2}}{n(n+1)\log(T)}\prod_{\begin{subarray}{c}3\leq
j\leq n\\\ j\text{ odd}\end{subarray}}\zeta(j)^{-1}$
$\displaystyle=(1+o_{\Omega}(1))\prod_{\begin{subarray}{c}3\leq j\leq n\\\
j\text{
odd}\end{subarray}}\zeta(j)^{-1}\int_{T\Omega}\frac{1}{\log^{+}(\det(x))}dx.$

Here we have applied Lemma 2.5 for the final equality. This completes the
proof of the following:

######  Lemma 7.15.

For $n\geq 3$ even, and $\Omega\subset\mathcal{O}_{n,0}$ a compact subset, we
have that as $T\rightarrow\infty$,

$\pi(T\Omega)=(1+o_{\Omega}(1))\prod_{\begin{subarray}{c}3\leq j\leq n\\\
j\text{
odd}\end{subarray}}\zeta(j)^{-1}\int_{T\Omega}\frac{1}{\log^{+}(\det(x))}dx.$

Together Theorem 4.1, Lemma 7.5, Lemma 7.7, Lemma 7.14, and Lemma 7.15 give us
Theorem 1.8.∎

##  8\. Final remarks

In this last section, we comment on variant problems that our methods can
handle. First, we note that we can instead count primes in arithmetic
progressions; for any $A>0$, $q\leq\log(T)^{A}$, and $(a,q)=1$, we can derive
an asymptotic for

$\\#\\{A\in\textrm{Mat}_{n}(\mathbb{Z}):\|A\|\leq T,\det(A)\equiv a\bmod
q,\det(A)\text{ is prime}\\}.$

This follows from applying Dirichlet’s theorem for primes in arithmetic
progressions in §3.2 rather than the prime number theorem.

Similarly, we can count almost-primes using the above techniques with a
slightly more involved computation with the Siegel masses, for instance
asymptotics for the following expression for any finite $k\in\mathbb{N}$:

$\\#\\{A\in\textrm{Mat}_{n}(\mathbb{Z}):\|A\|\leq T,\det(A)\text{ is a product
of $k$-primes}\\}.$

In comparison, for a general homogeneous polynomial $F(\mathbf{x})$ of degree
$d$ in $n$ variables, it is known that using the lower bound sieve of Diamond
and Halberstam, one can show that for a constant $C_{k}$ and for
$k>2d\cdot\frac{n-1}{n}+C_{k}$,

$\\#\\{\mathbf{x}\in\mathbb{Z}^{n}:\|\mathbf{x}\|\leq T,F(\mathbf{x})\text{ is
a product of at most $k$-primes}\\}\gg_{k}\frac{T^{n}}{\log(T)}.$

## Acknowledgments

The authors thank their advisor Peter Sarnak for suggesting the problem and
his insights described in his unpublished letter [31]. The authors would also
like to thank Anshul Adve, Valeriya Kovaleva, Hee Oh, Alina Ostafe, Igor
Shparlinski, and Nina Zubrilina for feedback on earlier drafts of the paper.
The second author is funded by the National Science Foundation Graduate
Research Fellowship Program under Grant No. DGE-2039656. Any opinions,
findings, and conclusions or recommendations expressed in this material are
those of the author(s) and do not necessarily reflect the views of the
National Science Foundation.

## References

  * [1] P. T. Bateman and R. A. Horn.  A heuristic asymptotic formula concerning the distribution of prime numbers.  Math. Comp., 16:363–367, 1962.
  * [2] A. Borel and Harish-Chandra.  Arithmetic subgroups of algebraic groups.  Ann. of Math. (2), 75:485–535, 1962.
  * [3] M. Borovoi and Z. Rudnick.  Hardy-Littlewood varieties and semisimple groups.  Invent. Math., 119(1):37–66, 1995.
  * [4] M. V. Borovoi.  The Hasse principle for homogeneous spaces.  J. Reine Angew. Math., 426:179–192, 1992.
  * [5] J. Bruedern and T. D. Wooley.  On waring’s problem for larger powers, 2022.
  * [6] J. W. S. Cassels.  Rational quadratic forms, volume 13 of London Mathematical Society Monographs.  Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London-New York, 1978.
  * [7] K. Conrad.  A multivariable hensel’s lemma.  Lecture note available at http://kconrad. math. uconn. edu/blurbs, 2020.
  * [8] J. H. Conway and N. J. A. Sloane.  Low-dimensional lattices. IV. The mass formula.  Proc. Roy. Soc. London Ser. A, 419(1857):259–286, 1988.
  * [9] S. Daniel.  On the divisor-sum problem for binary forms.  J. Reine Angew. Math., 507:107–129, 1999.
  * [10] K. Destagnol and E. Sofos.  Rational points and prime values of polynomials in moderately many variables.  Bull. Sci. Math., 156:102794, 33, 2019.
  * [11] W. Duke.  Hyperbolic distribution problems and half-integral weight Maass forms.  Invent. Math., 92(1):73–90, 1988.
  * [12] W. Duke, Z. Rudnick, and P. Sarnak.  Density of integer points on affine homogeneous varieties.  Duke Math. J., 71(1):143–179, 1993.
  * [13] M. Einsiedler, G. Margulis, A. Mohammadi, and A. Venkatesh.  Effective equidistribution and property ($\tau$).  Journal of the American Mathematical Society, 33(1):223–289, 2020.
  * [14] A. Eskin and H. Oh.  Representations of integers by an invariant polynomial and unipotent flows.  Duke Math. J., 135(3):481–506, 2006.
  * [15] J. Friedlander and H. Iwaniec.  Asymptotic sieve for primes.  Ann. of Math. (2), 148(3):1041–1065, 1998.
  * [16] J. Friedlander and H. Iwaniec.  The polynomial $X^{2}+Y^{4}$ captures its primes.  Ann. of Math. (2), 148(3):945–1040, 1998.
  * [17] J. Friedlander and H. Iwaniec.  Opera de cribro, volume 57 of American Mathematical Society Colloquium Publications.  American Mathematical Society, Providence, RI, 2010.
  * [18] D. R. Heath-Brown.  Primes represented by $x^{3}+2y^{3}$.  Acta Math., 186(1):1–84, 2001.
  * [19] D. R. Heath-Brown and B. Z. Moroz.  On the representation of primes by cubic polynomials in two variables.  Proc. London Math. Soc. (3), 88(2):289–312, 2004.
  * [20] H. Iwaniec.  Fourier coefficients of modular forms of half-integral weight.  Invent. Math., 87(2):385–401, 1987.
  * [21] Y. Kitaoka.  Two theorems on the class number of positive definite quadratic forms.  Nagoya Mathematical Journal, 51:79–89, 1973.
  * [22] V. Kovaleva.  On the distribution of equivalence classes of random symmetric $p$-adic matrices.  Mathematika, 69(4):903–933, 2023.
  * [23] S. Lang and A. Weil.  Number of points of varieties in finite fields.  American Journal of Mathematics, 76(4):819–827, 1954.
  * [24] J. V. Linnik and M. Keane.  Ergodic properties of algebraic fields, volume 45.  Springer, 1968.
  * [25] Y. V. Linnik and B. F. Skubenko.  On the asymptotic behavior of integral matrices of third order.  In Doklady Akademii Nauk, volume 146, pages 1007–1008. Russian Academy of Sciences, 1962.
  * [26] J. MacWilliams.  Orthogonal matrices over finite fields.  Amer. Math. Monthly, 76:152–164, 1969.
  * [27] J. Maynard.  Primes represented by incomplete norm forms.  Forum Math. Pi, 8:e3, 128, 2020.
  * [28] H. Oh.  Hardy-Littlewood system and representations of integers by an invariant polynomial.  Geom. Funct. Anal., 14(4):791–809, 2004.
  * [29] M. Ratner.  On raghunathan’s measure conjecture.  Annals of Mathematics, 134(3):545–607, 1991.
  * [30] P. Sarnak.  Diophantine problems and linear groups.  In Proceedings of the International Congress of Mathematicians, volume 1, pages 459–471, 1990.
  * [31] P. Sarnak.  Letter to Sarah and Valeriya, Jan 2022.
  * [32] C. L. Siegel.  Über die Analytische Theorie der Quadratischen Formen.  Ann. Math., 36:527–606, 1935.
  * [33] A. Weil.  Adeles and algebraic groups, volume 23 of Progress in Mathematics.  Birkhäuser, Boston, MA, 1982.  With appendices by M. Demazure and Takashi Ono.
  * [34] L. Zhang.  Volumes of orthogonal groups and unitary groups.  arXiv: Mathematical Physics, 2015.