# Sharp lower bounds for moments of $\zeta^{\prime}(\rho)$

Peng Gao  School of Mathematical Sciences, Beihang University, Beijing 100191,
P. R. China<EMAIL_ADDRESS>
###### Abstract.

We study the $2k$-th discrete moment of the derivative of the Riemann zeta-
function at nontrivial zeros to establish sharp lower bounds for all real
$k\geq 0$ under the Riemann hypothesis (RH).

Mathematics Subject Classification (2010): 11M06, 11M26

Keywords: lower bounds, moments, nontrivial zeros, Riemman zeta-function

##  1\. Introduction

Various types of moments of the Riemann zeta function $\zeta(s)$ have been
extensively studied in the literature. In this paper, we are interested in the
$2k$-th discrete moment of the derivative of $\zeta(s)$ at nontrivial zeros
denoted by

$\displaystyle J_{k}(T):=\frac{1}{N(T)}\sum_{0<\Im(\rho)\leq
T}|\zeta^{\prime}(\rho)|^{2k},$

where we write $\rho$ for the nontrivial zeros of $\zeta(s)$ and

$\displaystyle N(T)=\sum_{0<\Im(\rho)\leq T}1.$

In [Gonek], S. M. Gonek initiated the study on $J_{k}(T)$ to show that under
the truth of the Riemann hypothesis (RH), one has asymptotically

$\displaystyle J_{1}(T)\sim\frac{1}{12}(\log T)^{3}.$

Regarding the order of magnitude for $J_{k}(T)$, S. M. Gonek [Gonek1] and D.
Hejhal [Hejhal] independently conjectured that for any real $k$,

(1.1) $\displaystyle J_{k}(T)\asymp(\log T)^{k(k+2)}.$

A precisely asymptotic formula is further conjectured by C. P. Hughes, J. P.
Keating and N. O’Connell in [HKO], building on connections with the random
matrix theory. The evidence from the random matrix side also suggests that
(1.1) may not be valid for $k\leq-3/2$.

A proof of (1.1) for the case of $k=2$ is given by N. Ng [Ng] assuming RH.
Under RH and the additional assumption that the zeros of $\zeta(s)$ are
simple, S. M. Gonek [Gonek1] obtained a lower bound for $J_{-1}(T)$ of the
conjectured order of magnitude. An explicit estimation for the constant
involved is further given by M. B. Milinovich and N. Ng [MN]. In [MN1], M. B.
Milinovich and N. Ng also proved that for all natural number $k$,

(1.2) $\displaystyle J_{k}(T)\gg_{k}(\log T)^{k(k+2)}.$

On the other hand, M. B. Milinovich [Milinovich] proved that under RH, for any
natural number $k$ and any $\varepsilon>0$,

(1.3) $\displaystyle J_{k}(T)\ll_{k,\varepsilon}(\log
T)^{k(k+2)+\varepsilon}.$

In [Kirila], S. Kirila obtained sharp upper bounds for $J_{k}(T)$ for all real
$k>0$ under RH. In particular, this implies the validity of (1.3) for natural
numbers $k$ without the extra $\varepsilon$ power. Together with (1.2), we see
that (1.1) is valid for all natural numbers $k$.

The approach taken by M. B. Milinovich and N. Ng to obtain their result
concerning (1.2) follows from a simple and powerful method developed by Z.
Rudnick and K. Soundararajan in [R&Sound, R&Sound1] for establishing lower
bounds for moments of families of $L$-functions, while the approach taken by
S. Kirila follows from a method of K. Soundararajan [Sound01] together with
its refinement by A. J. Harper [Harper] for establishing upper bounds for
moments of families of $L$-functions.

There are now a few more approaches towards establishing sharp bounds for
moments of $L$-functions, notably the upper bounds principal of M. Radziwiłł
and K. Soundararajan [Radziwill&Sound] and the lower bounds principal of W.
Heap and K. Soundararajan [H&Sound]. One then expects to apply these
approaches to obtain sharp bounds concerning $J_{k}(T)$. In fact, it is
pointed out in [Kirila] that one should be able to establish sharp lower
bounds for all real $k>0$ using the approaches in [Radziwill&Sound,
Radziwill&Sound1]. The aim of this paper is to achieve this and our main
result is as follows.

######  Theorem 1.1.

Assuming RH. For large $T$ and any $k\geq 0$, we have

$\displaystyle J_{k}(T)\gg_{k}(\log T)^{k(k+2)}.$

We shall instead apply the lower bounds principal of W. Heap and K.
Soundararajan [H&Sound] in the proof of Theorem 1.1. The proof also uses the
arguments by A. J. Harper in [Harper] and by S. Kirila in [Kirila]. Combining
Theorem 1.1 and the above mentioned result of S. Kirila in [Kirila], we
immediately obtain the following result concerning the order of magnitude of
$J_{k}(T)$.

######  Corollary 1.2.

Assuming RH. For large $T$, the estimation given in (1.1) is valid for all
real $k\geq 0$.

##  2\. Preliminaries

We now introduce some notations and auxiliary results used in the paper. We
assume the truth of RH throughout so that we may write each nontrivial zero
$\rho$ of $\zeta(s)$ as $\rho=\tfrac{1}{2}+i\gamma$, where we denote
$\gamma\in\mathbb{R}$ for the imaginary part of $\rho$. We denote $\omega(n)$
for the number of distinct prime factors of $n$ and $\Omega(n)$ for the number
of prime powers dividing $n$. We note the following estimation (see [MVa,
Theorem 2.10]) for $\omega(n)$,

(2.1) $\displaystyle\omega(n)\leq\frac{\log n}{\log\log
n}(1+O(\frac{1}{\log\log n})),\quad n\geq 3.$

We also define $\Lambda_{j}(n)$ for all integers $j\geq 0$ to be the
coefficient of $n^{-s}$ in the Dirichlet series expansion of
$(-1)^{j}\zeta^{(j)}(s)/\zeta(s)$. In particular, we have
$\Lambda_{1}(n)=\Lambda(n)$, the usual von Mangoldt function. We extend the
definition of $\Lambda$ to all real numbers $x$ by defining $\Lambda(x)=0$
when $x$ is not an integer and we note the following uniform version of
Landau’s formula [Landau1912], originally proved by S. M. Gonek [Gonek93].

######  Lemma 2.1.

Assume RH. Then we have for $T$ large and any positive integers $a,b$,

(2.2) $\displaystyle\begin{split}\sum_{T<\gamma\leq
2T}(a/b)^{i\gamma}=\begin{cases}N(T,2T),\quad a=b,\\\
\displaystyle-\frac{T}{2\pi}\frac{\Lambda(a/b)}{\sqrt{a/b}}+O\big{(}\sqrt{ab}(\log
T)^{2}\big{)},\quad a>b,\\\
\displaystyle-\frac{T}{2\pi}\frac{\Lambda(b/a)}{\sqrt{b/a}}+O\big{(}\sqrt{ab}(\log
T)^{2}\big{)},\quad b>a,\end{cases}\end{split}$

where we denote $N(T,2T)$ for the number of nontrivial zeros $\rho$ such that
$T<\Im(\rho)\leq 2T$.

The cases when $a\neq b$ of Lemma 2.1 are given in [Kirila, Lemma 5.1] while
the case $a=b$ of Lemma 2.1 is trivial. Recall that the Riemann–von Mangoldt
formula asserts (see [Da, Chapter 15]) that

$\displaystyle N(T)=\frac{T}{2\pi}\log\frac{T}{2\pi e}+O(\log T).$

It follows from this and the relation $N(T,2T)=N(2T)-N(T)$ that

(2.3) $\displaystyle N(T,2T)\ll T\log T.$

We reserve the letter $p$ for a prime number in this paper and we note that
$\Lambda_{2}(n)$ is supported on integers $n$ with $\omega(n)\leq 2$
satisfying for primes $p,q,p\neq q$ and positive integers $i,j$,

(2.4) $\displaystyle\Lambda_{2}(p^{i})\ll i(\log
p)^{2},\quad\Lambda_{2}(p^{i}q^{j})\ll(\log p)(\log q).$

We recall the following well-known Mertens’ formula (see [MVa1, Theorem 2.7]).

######  Lemma 2.2.

Let $x\geq 2$. We have, for some constant $b$,

$\sum_{p\leq x}\frac{1}{p}=\log\log x+b+O\Big{(}\frac{1}{\log x}\Big{)}.$

We end this section by including a mean value theorem given in [MN1, Lemma
4.1] concerning integrals over Dirichlet polynomials.

######  Lemma 2.3.

Let $\\{a_{n}\\}$ and $\\{b_{n}\\}$ be sequences of complex numbers. Let
$T_{1}$ and $T_{2}$ be positive real numbers and $g(t)$ be a real-valued
function that is continuously differentiable on the interval $[T_{1},T_{2}]$.
Then

$\displaystyle\begin{split}&\int^{T_{2}}_{T_{1}}g(t)\left(\sum^{\infty}_{n=1}a_{n}n^{-it}\right)\left(\sum^{\infty}_{n=1}b_{n}n^{it}\right)dt\\\
=&\int^{T_{2}}_{T_{1}}g(t)dt\sum^{\infty}_{n=1}a_{n}b_{n}+O\left(\left(|g(T_{1})|+|g(T_{2})|+\int^{T_{2}}_{T_{1}}|g^{\prime}(t)|dt\right)\left(\sum^{\infty}_{n=1}n|a_{n}|^{2}\right)^{1/2}\left(\sum^{\infty}_{n=1}n|b_{n}|^{2}\right)^{1/2}\right).\end{split}$

##  3\. Proof of Theorem 1.1

###  3.1. The lower bound principle

We assume that $T$ is a large number throughout the proof. As the case $k=0$
is trivial, we only consider the case $k>0$ in the proof. Moreover, we note
that in the rest of the paper, the explicit constants involved in estimations
using $\ll$ or the big-$O$ notations depend on $k$ only and are uniform with
respect to $\rho$. We further make the convention that an empty product is
defined to be $1$.

We follow the ideas of A. J. Harper in [Harper] and the notations of S. Kirila
in [Kirila] to define for a large number $M$ depending on $k$ only,

$\alpha_{0}=0,\;\;\;\;\;\alpha_{j}=\frac{20^{j-1}}{(\log\log
T)^{2}}\;\;\;\forall\;j\geq
1,\quad\mathcal{J}=\mathcal{I}_{k,T}=1+\max\\{j:\alpha_{j}\leq 10^{-M}\\}.$

It follows from the above notations and Lemma 2.2 that we have for $1\leq
j\leq\mathcal{J}-1$ and $T$ large enough,

(3.1) $\displaystyle\sum_{T^{\alpha_{j}}<p\leq
T^{\alpha_{j+1}}}\frac{1}{p}=\log\alpha_{j+1}-\log\alpha_{j}+o(1)=\log
20+o(1)\leq 10.$

We denote for any real number $\ell$ and any $x\in\mathbb{R}$,

$\displaystyle E_{\ell}(x)=\sum_{j=0}^{\lceil\ell\rceil}\frac{x^{j}}{j!}.$

We then define for any real number $\alpha$ and any $1\leq j\leq\mathcal{J}$,

$\displaystyle{\mathcal{P}}_{j}(s)=$ $\displaystyle\sum_{p\in
I_{j}}\frac{1}{p^{s}},\quad{\mathcal{N}}_{j}(s,\alpha)=E_{e^{2}k\alpha^{-3/4}_{j}}\Big{(}\alpha{\mathcal{P}}_{j}(s)\Big{)},\quad{\mathcal{N}}(s,\alpha)=\prod^{\mathcal{J}}_{j=1}{\mathcal{N}}_{j}(s,\alpha).$

Denote $g(n)$ for the multiplicative function given on prime powers by
$g(p^{r})=1/r!$ and define functions $b_{j}(n),1\leq j\leq{\mathcal{J}}$ such
that $b_{j}(n)=0$ or $1$ and that $b_{j}(n)=1$ only when $n$ is composed of at
most $\lceil e^{2}k\alpha^{-3/4}_{j}\rceil$ primes, all from the interval
$I_{j}$. We then have

$\displaystyle{\mathcal{N}}_{j}(s,\alpha)=\sum_{n_{j}}\frac{\alpha^{\Omega(n_{j})}}{g(n_{j})}b_{j}(n_{j})\frac{1}{n^{s}_{j}},\quad
1\leq j\leq{\mathcal{J}}.$

Note that each ${\mathcal{N}}_{j}(s,\alpha)$ is a short Dirichlet polynomial
of length at most $T^{\alpha_{j}\lceil e^{2}k\alpha^{-3/4}_{j}\rceil}$. By
taking $T$ large enough, we notice that

$\displaystyle\sum^{\mathcal{J}}_{j=1}\alpha_{j}\lceil
e^{2}k\alpha^{-3/4}_{j}\rceil\leq 40e^{2}k10^{-M/4}.$

It follows that ${\mathcal{N}}(s,\alpha)$ is also a short Dirichlet polynomial
of length at most $T^{40e^{2}k10^{-M/4}}$.

Moreover, we write for simplicity that

(3.2)
$\displaystyle{\mathcal{N}}(s,\alpha)=\sum_{n}\frac{a_{\alpha}(n)}{n^{s}}.$

We note that $a_{\alpha}(n)\neq 0$ only when $n=\prod_{1\leq
j\leq\mathcal{J}}n_{j}$, in which case we have

$\displaystyle
a_{\alpha}(n)=\prod_{n_{j}}\frac{\alpha^{\Omega(n_{j})}}{g(n_{j})}b_{j}(n_{j}).$

Taking note also the estimation that for all integers $i\geq 0$,

$\displaystyle\frac{\alpha^{i}}{i!}\ll e^{|\alpha|},$

we conclude from above discussions that for all $n$,

(3.3) $\displaystyle a_{\alpha}(n)\leq e^{|\alpha|\omega(n)},\quad
a_{k}(n)=0,\text{ when }n>T^{40e^{2}k10^{-M/4}}.$

We apply the above estimations and (2.1) in (3.2) to see that for
$\Re(s)\geq-1/\log T$ and $T$ large enough,

(3.4) $\displaystyle|{\mathcal{N}}(s,\alpha)|\ll e^{|\alpha|\frac{\log
T}{\log\log T}(1+O(\frac{1}{\log\log T}))}T^{40e^{2}k10^{-M/4}(1+1/\log T)}.$

In the proof of Theorem 1.1, we need the following bounds concerning
expressions involving with various ${\mathcal{N}}(s,\alpha)$ .

######  Lemma 3.2.

With the notations as above, we have for $0<k\leq 1/2$ and $1\leq
j\leq\mathcal{J}$,

(3.5)
$\displaystyle\begin{split}|\mathcal{N}_{j}(s,k)|^{2/k}|\mathcal{N}_{j}(s,k-1)|^{2}\leq|{\mathcal{N}}_{j}(s,k)|^{2}\left(1+e^{-e^{2}k\alpha^{-3/4}_{j}}\right)^{2/k+2}\left(1-e^{-e^{2}k\alpha^{-3/4}_{j}}\right)^{-2}+|{\mathcal{Q}}_{j}(s,k)|^{2r_{k}}.\end{split}$

We also have for $k>1/2$ and $1\leq j\leq\mathcal{J}$,

(3.6)
$\displaystyle\begin{split}|\mathcal{N}_{j}(s,k-1)\mathcal{N}_{j}(s,k)|^{\frac{2k}{2k-1}}\leq|{\mathcal{N}}_{j}(s,k)|^{2}\left(1+e^{-e^{2}k\alpha^{-3/4}_{j}}\right)^{\frac{2k}{2k-1}}\left(1-e^{-e^{2}k\alpha^{-3/4}_{j}}\right)^{-2}+|{\mathcal{Q}}_{j}(s,k)|^{2r_{k}}.\end{split}$

Here the implied constants in (3.5) and (3.6) are absolute, and we define

${\mathcal{Q}}_{j}(s,k)=\Big{(}\frac{64\max(2,k+3/2){\mathcal{P}}_{j}(s)}{\lceil
e^{2}k\alpha^{-3/4}_{j}\rceil}\Big{)}^{\lceil e^{2}k\alpha^{-3/4}_{j}\rceil},$

with $r_{k}=2+\lceil 1/k\rceil$ for $0<k\leq 1/2$ and $r_{k}=1+\lceil
2k/(2k-1)\rceil$ for $k>1/2$.

###### Proof.

As in the proof of [Gao2021-3, Lemma 3.4], we have for $|z|\leq aK/20$ with
$0<a\leq 2$,

(3.7)
$\displaystyle\Big{|}\sum_{r=0}^{K}\frac{z^{r}}{r!}-e^{z}\Big{|}\leq\frac{|z|^{K}}{K!}\leq\Big{(}\frac{ae}{20}\Big{)}^{K}.$

By taking $z=\alpha{\mathcal{P}}_{j}(s),K=\lceil
e^{2}k\alpha^{-3/4}_{j}\rceil$ and $a=\min(|\alpha|,2)$ in (3.7), we see that
when $|{\mathcal{P}}_{j}(s)|\leq\lceil
e^{2}k\alpha^{-3/4}_{j}\rceil/(20(1+|\alpha|))$,

$\displaystyle{\mathcal{N}}_{j}(s,\alpha)\leq$
$\displaystyle\exp(\alpha{\mathcal{P}}_{j}(s))\left(1+\exp(|\alpha{\mathcal{P}}_{j}(s)|)\left(\frac{ae}{20}\right)^{e^{2}k\alpha^{-3/4}_{j}}\right)\leq\exp(\alpha{\mathcal{P}}_{j}(s))\left(1+e^{-e^{2}k\alpha^{-3/4}_{j}}\right).$

Similarly, we have

$\displaystyle{\mathcal{N}}_{(}s,\alpha)\geq$
$\displaystyle\exp(\alpha{\mathcal{P}}_{j}(s))\left(1-\exp(|\alpha{\mathcal{P}}_{j}(s)|)\left(\frac{ae}{20}\right)^{e^{2}k\alpha^{-3/4}_{j}}\right)\geq\exp(\alpha{\mathcal{P}}_{j}(s))\left(1-e^{-e^{2}k\alpha^{-3/4}_{j}}\right).$

We apply the above estimation to
${\mathcal{N}}_{j}(s,k),{\mathcal{N}}_{j}(s,k-1)$ to see that when $0<k\leq
1/2$ and $|{\mathcal{P}}_{j}(s)|\leq\lceil e^{2}k\alpha^{-3/4}_{j}\rceil/60$,
we have

(3.8)
$\displaystyle\begin{split}|{\mathcal{N}}_{j}(s,k)|^{\frac{2}{k}}|{\mathcal{N}}_{j}(s,k-1)|^{2}\leq&\exp(2k\Re{\mathcal{P}}_{j}(s))\left(1+e^{-e^{2}k\alpha^{-3/4}_{j}}\right)^{2/k+2}\\\
\leq&|{\mathcal{N}}_{j}(s,k)|^{2}\left(1+e^{-e^{2}k\alpha^{-3/4}_{j}}\right)^{2/k+2}\left(1-e^{-e^{2}k\alpha^{-3/4}_{j}}\right)^{-2}.\end{split}$

The above arguments also imply that when $k>1/2$ and
$|{\mathcal{P}}_{j}(s)|\leq\lceil e^{2}k\alpha^{-3/4}_{j}\rceil/60$, then

(3.9)
$\displaystyle\begin{split}|\mathcal{N}_{j}(s,2k-1)|^{\frac{2k}{2k-1}}\leq|{\mathcal{N}}_{j}(s,k)|^{2}\left(1+e^{-e^{2}k\alpha^{-3/4}_{j}}\right)^{\frac{2k}{2k-1}}\left(1-e^{-e^{2}k\alpha^{-3/4}_{j}}\right)^{-2}.\end{split}$

On the other hand, when $|{\mathcal{P}}_{j}(s)|\geq\lceil
e^{2}k\alpha^{-3/4}_{j}\rceil/60$, we have that

(3.10)
$\displaystyle\begin{split}{\mathcal{N}}_{j}(s,\alpha)\leq\sum_{r=0}^{\lceil
e^{2}k\alpha^{-3/4}_{j}\rceil}\frac{|\alpha{\mathcal{P}}_{j}(s)|^{r}}{r!}&\leq|(|\alpha|+1){\mathcal{P}}_{j}(s)|^{\lceil
e^{2}k\alpha^{-3/4}_{j}\rceil}\sum_{r=0}^{\lceil
e^{2}k\alpha^{-3/4}_{j}\rceil}\Big{(}\frac{60}{\lceil
e^{2}k\alpha^{-3/4}_{j}\rceil}\Big{)}^{\lceil
e^{2}k\alpha^{-3/4}_{j}\rceil-r}\frac{1}{r!}\\\
&\leq\Big{(}\frac{64(|\alpha|+1)|{\mathcal{P}}_{j}(s)|}{\lceil
e^{2}k\alpha^{-3/4}_{j}\rceil}\Big{)}^{\lceil
e^{2}k\alpha^{-3/4}_{j}\rceil}.\end{split}$

We then set $\alpha=1$ in the last expression in (3.10) to deduce that when
$0<k\leq 1/2$ and $|{\mathcal{P}}_{j}(s)|\geq\lceil
e^{2}k\alpha^{-3/4}_{j}\rceil/60$,

(3.11)
$\displaystyle\begin{split}|{\mathcal{N}}_{j}(s,k)|^{\frac{2}{k}}|{\mathcal{N}}_{j}(s,k-1)|^{2}\leq|{\mathcal{Q}}_{j}(s,k)|^{2r_{k}}.\end{split}$

Moreover, we set $\alpha=k$ in the last expression in (3.10) to deduce that
when $k>1/2$ and $|{\mathcal{P}}_{j}(s)|\geq\lceil
e^{2}k\alpha^{-3/4}_{j}\rceil/(10(1+|\alpha|))$,

(3.12)
$\displaystyle\begin{split}|{\mathcal{N}}_{j}(s,k-1){\mathcal{N}}_{j}(s,k)|^{\frac{2k}{2k-1}}\leq|{\mathcal{Q}}_{j}(s,k)|^{2r_{k}}.\end{split}$

The assertion of the lemma now follows from (3.8), (3.9), (3.11) and (3.12). ∎

Next, we state the needed lower bounds principle of W. Heap and K.
Soundararajan in [H&Sound] for our situation.

######  Lemma 3.3.

With notations as above. For $0<k\leq 1/2$, we have

(3.13) $\displaystyle\begin{split}\sum_{0<\gamma\leq
T}\zeta^{\prime}(\rho)\mathcal{N}(\rho,k-1)\mathcal{N}(\overline{\rho},k)\ll&\Big{(}\sum_{0<\gamma\leq
T}|\zeta^{\prime}(\rho)|^{2k}\Big{)}^{1/2}\Big{(}\sum_{0<\gamma\leq
T}|\zeta^{\prime}(\rho)|^{2}|\mathcal{N}(\rho,k-1)|^{2}\Big{)}^{(1-k)/2}\\\
&\times\Big{(}\sum_{0<\gamma\leq
T}\prod^{\mathcal{J}}_{i=1}\big{(}|{\mathcal{N}}_{i}(\rho,k)|^{2}+|{\mathcal{Q}}_{i}(\rho,k)|^{2r_{k}}\big{)}\Big{)}^{k/2}.\end{split}$

For $k>1/2$, we have

(3.14) $\displaystyle\begin{split}&\sum_{0<\gamma\leq
T}\zeta^{\prime}(\rho)\mathcal{N}(\rho,k-1)\mathcal{N}(\overline{\rho},k)\leq\Big{(}\sum_{0<\gamma\leq
T}|\zeta^{\prime}(\rho)|^{2k}\Big{)}^{\frac{1}{2k}}\Big{(}\sum_{0<\gamma\leq
T}\prod^{\mathcal{J}}_{j=1}\big{(}|{\mathcal{N}}_{j}(\rho,k)|^{2}+|{\mathcal{Q}}_{j}(\rho,k)|^{2r_{k}}\big{)}\Big{)}^{\frac{2k-1}{2k}}.\end{split}$

The implied constants in (3.13) and (3.14) depend on $k$ only.

###### Proof.

We assume $0<k\leq 1/2$ first and apply Hölder’s inequality to see that the
left side of (3.13) is

(3.15) $\displaystyle\begin{split}\leq&\Big{(}\sum_{0<\gamma\leq
T}|\zeta^{\prime}(\rho)|^{2k}\Big{)}^{1/2}\Big{(}\sum_{0<\gamma\leq
T}|\zeta^{\prime}(\rho)|^{2}|\mathcal{N}(\rho,k-1)|^{2}\Big{)}^{(1-k)/2}\Big{(}\sum_{0<\gamma\leq
T}|\mathcal{N}(\rho,k)|^{2/k}|\mathcal{N}(\rho,k-1)|^{2}\Big{)}^{k/2}.\end{split}$

We apply the estimation in (3.5) in the last sum of (3.15) above and note
that, upon applying the estimation that $1-e^{-u}\geq e^{-1/u}$ for $u>0$, we
have

$\displaystyle\prod_{1\leq
j\leq{\mathcal{J}}}\left(1+e^{-e^{2}k\alpha^{-3/4}_{j}}\right)^{2/k+2}\left(1-e^{-e^{2}k\alpha^{-3/4}_{j}}\right)^{-2}\leq$
$\displaystyle\prod_{1\leq
j\leq{\mathcal{J}}}\left(1-e^{-e^{2}k\alpha^{-3/4}_{j}}\right)^{-2}\leq\prod_{1\leq
j\leq{\mathcal{J}}}e^{2\alpha^{3/4}_{j}/(e^{2}k)}<\infty.$

This leads to the estimation given in (3.13).

It remains to consider the case $k>1/2$ and we apply Hölder’s inequality again
to see that the left side of (3.14) is

(3.16) $\displaystyle\begin{split}\leq\Big{(}\sum_{0<\gamma\leq
T}|\zeta^{\prime}(\rho)|^{2k}\Big{)}^{\frac{1}{2k}}\Big{(}\sum_{0<\gamma\leq
T}|\mathcal{N}(\rho,k-1)\mathcal{N}(\rho,k)|^{\frac{2k}{2k-1}}\Big{)}^{\frac{2k-1}{2k}}.\end{split}$

We apply the estimation in (3.6) in the last sum of (3.16) above and note that
the product

$\prod_{1\leq
j\leq{\mathcal{J}}}\left(1+e^{-e^{2}k\alpha^{-3/4}_{j}}\right)^{\frac{2k}{2k-1}}\left(1-e^{-e^{2}k\alpha^{-3/4}_{j}}\right)^{-2}<\infty.$

This leads to the estimation given in (3.14) and this completes the proof. ∎

It follows from the above lemma and the observation that
$\overline{\rho}=1-\rho$ that in order to establish Theorem 1.1, it suffices
to prove the following three propositions.

######  Proposition 3.4.

With notations as above, we have for $k>0$,

(3.17) $\displaystyle\sum_{0<\gamma\leq
T}\zeta^{\prime}(\rho)\mathcal{N}(\rho,k-1)\mathcal{N}(1-\rho,k)\gg T(\log
T)^{k^{2}+2}.$

######  Proposition 3.5.

With notations as above, we have for $k>0$,

$\displaystyle\sum_{0<\gamma\leq
T}\prod^{\mathcal{J}}_{j=1}\big{(}|{\mathcal{N}}_{j}(\rho,k)|^{2}+|{\mathcal{Q}}_{j}(\rho,k)|^{2r_{k}}\big{)}\ll$
$\displaystyle T(\log T)^{k^{2}+1}.$

######  Proposition 3.6.

With notations as above, we have for $0<k\leq 1/2$,

$\displaystyle\sum_{0<\gamma\leq
T}|\zeta^{\prime}(\rho)|^{2}|\mathcal{N}(\rho,k-1)|^{2}\ll T(\log
T)^{k^{2}+3}.$

We shall prove the above propositions in the rest of the paper.

###  3.7. Proof of Proposition 3.4

The proof follows largely the arguments in Section 5 of [MN1]. We begin by
recalling a few results on the Riemann zeta function $\zeta(s)$. Notice that
$\zeta(s)$ satisfies the functional equation (see [MVa, Corollary 10.4]):

(3.18) $\displaystyle\zeta(s)=\chi(s)\zeta(1-s),$

where

$\chi(s)=2^{s}\pi^{s-1}\Gamma(1-s)\sin(\pi s/2)\ .$

Logarithmically differentiating the functional equation above implies that

(3.19)
$\displaystyle\frac{\zeta^{\prime}}{\zeta}(1-s)=\frac{\chi^{\prime}}{\chi}(s)-\frac{\zeta^{\prime}}{\zeta}(s).$

Moreover, we have (see [MN1, (8)]) uniformly for $-1\leq\sigma\leq 2$ and
$|t|\geq 1$,

(3.20)
$\displaystyle\frac{\chi^{\prime}}{\chi}(\sigma+it)=\frac{\chi^{\prime}}{\chi}(1-\sigma-
it)=-\log\left(\frac{|t|}{2\pi}\right)+O(\frac{1}{|t|}).$

Note that it follows from [Da, p. 108] that for every $t\geq 2$ and all
nontrivial zeros $\rho=\tfrac{1}{2}+i\gamma$ of $\zeta(s)$, there exists a
number $T$ satisfying $t\leq T\leq t+1$ such that

(3.21) $\displaystyle\frac{\zeta^{\prime}}{\zeta}(\sigma+it)\ll(\log
T)^{2},\quad\text{for}-1\leq\sigma\leq 2\ \ \text{and}\ \ |\gamma-T|\gg(\log
T)^{-1}.$

Note also that by differentiating the functional equation (3.18) above, we
have

(3.22)
$\displaystyle\zeta^{\prime}(s)=-\chi(s)\left(\zeta^{\prime}(1-s)-\frac{\chi^{\prime}}{\chi}(s)\zeta(1-s)\right).$

We deduce from this that if we denote the left side expression in (3.17) by
$S$, then

$\displaystyle S$ $\displaystyle=-\sum_{0<\gamma\leq
T}\chi(\rho)\zeta^{\prime}(1-\rho)\mathcal{N}(\rho,k-1)\mathcal{N}(1-\rho,k)=\frac{1}{2\pi
i}\int_{\mathcal{C}}\frac{\zeta^{\prime}}{\zeta}(1-s)\chi(s)\zeta^{\prime}(1-s)\mathcal{N}(s,k-1)\mathcal{N}(1-s,k)\,ds,$

where $\mathcal{C}$ is the positively oriented rectangle with vertices at
$1-\kappa+i,\kappa+i,\kappa+iT,$ and $1-\kappa+iT$, and $\kappa=1+(\log
T)^{-1}$. Here we may choose $T$ to satisfy the conditions in (3.21). We then
apply (3.4), (3.21) and the estimations

$\begin{split}\chi(s)\ll T^{1/2-\sigma},\quad\zeta^{\prime}(1-s)\ll
T^{\sigma/2+\epsilon}\ \end{split}$

to see that the integral is bounded by $O(T^{1-\varepsilon})$ on the
horizontal edges of the contour.

Next, we deduce from (3.22) and the functional equation (3.18) that

$\chi(s)\zeta^{\prime}(1-s)=-\zeta^{\prime}(s)+\frac{\chi^{\prime}}{\chi}(s)\zeta(s).$

Combining this with (3.19), we see that the integral on the right edge of the
contour equals

$S_{R}=\frac{1}{2\pi
i}\int_{\kappa+i}^{\kappa+iT}\left(\frac{\chi^{\prime}}{\chi}(s)^{2}\zeta(s)-2\frac{\chi^{\prime}}{\chi}(s)\zeta^{\prime}(s)+\frac{\zeta^{\prime}}{\zeta}(s)\zeta^{\prime}(s)\right)\mathcal{N}(s,k-1)\mathcal{N}(1-s,k)\,ds.$

Also, the integral on the left edge of the contour equals

$S_{L}=\frac{1}{2\pi
i}\int_{1-\kappa+iT}^{1-\kappa+i}\frac{\zeta^{\prime}}{\zeta}(1-s)\chi(s)\zeta^{\prime}(1-s)\mathcal{N}(s,k-1)\mathcal{N}(1-s,k)\,ds.$

we make a change of variable $s\to 1-s$ to see that $S_{L}=-\overline{I}_{L}$,
where

$I_{L}=\frac{1}{2\pi
i}\int_{\kappa+i}^{\kappa+iT}\chi(1-s)\frac{\zeta^{\prime}}{\zeta}(s)\zeta^{\prime}(s)\mathcal{N}(1-s,k-1)\mathcal{N}(s,k)\,ds.$

We then conclude that

$S=S_{R}-\overline{I}_{L}+O(T^{1-\varepsilon}).$

We now apply (3.4), (3.20) and the bounds (see [MVa, Corollary 1.17, Theorem
6.7]) that when $s=\kappa+it$,

$\displaystyle\zeta(s)\ll$
$\displaystyle(1+(|t|+4)^{1-\kappa})\min(\frac{1}{|\kappa-1|},\log(|t|+4))+\frac{1}{s-1}+O(1),$
$\displaystyle\frac{\zeta^{\prime}}{\zeta}(s)\ll$
$\displaystyle\log(|t|+4)-\frac{1}{s-1}+O(1),$

to see that

(3.23) $\displaystyle S_{R}=\frac{1}{2\pi
i}\int_{\kappa+i}^{\kappa+iT}\left(\log^{2}\left(\frac{t}{2\pi}\right)\zeta(s)+2\log\left(\frac{t}{2\pi}\right)\zeta^{\prime}(s)+\frac{\zeta^{\prime}}{\zeta}(s)\zeta^{\prime}(s)\right)\mathcal{N}(s,k-1)\mathcal{N}(1-s,k)\,ds+O(T^{1-\varepsilon}).$

To evaluate $S_{R}$, we define the Dirichlet convolution $f*g$ for two
arithmetic functions $f(k),g(k)$ by

$\displaystyle f*g(k)=\sum_{mn=k}f(m)g(n).$

We then denote the integral given in (3.23) as a sum of three terms:
$S_{R,1},S_{R,2}$ and $S_{R,3}$, where

$\displaystyle S_{R,1}=$ $\displaystyle\frac{1}{2\pi
i}\int_{\kappa+i}^{\kappa+iT}\log^{2}\left(\frac{t}{2\pi}\right)\zeta(s)\mathcal{N}(s,k-1)\mathcal{N}(1-s,k)\,ds,$
$\displaystyle S_{R,2}=$ $\displaystyle\frac{2}{2\pi
i}\int_{\kappa+i}^{\kappa+iT}\log\left(\frac{t}{2\pi}\right)\zeta^{\prime}(s)\mathcal{N}(s,k-1)\mathcal{N}(1-s,k)\,ds,$
$\displaystyle S_{R,3}=$ $\displaystyle\frac{1}{2\pi
i}\int_{\kappa+i}^{\kappa+iT}\frac{\zeta^{\prime}}{\zeta}(s)\zeta^{\prime}(s)\mathcal{N}(s,k-1)\mathcal{N}(1-s,k)\,ds.$

We use the notation given in (3.2) and apply Lemma 2.3 to evaluate $S_{R,1}$
to obtain that

$\displaystyle S_{R,1}=$
$\displaystyle\left(\frac{1}{2\pi}\int_{1}^{T}\left(\log^{2}\frac{t}{2\pi}\right)dt\right)\sum_{n}\frac{1*a_{k-1}(n)\cdot
a_{k}(n)}{n}$
$\displaystyle+\left(\left(\log^{2}T+\int^{T}_{1}|(\log^{2}\frac{t}{2\pi})^{\prime}|dt\right)\left(\sum^{\infty}_{n=1}\frac{(1*a_{k-1})(n)^{2}}{n^{2\kappa-1}}\right)^{\tfrac{1}{2}}\left(\sum^{\infty}_{n=1}\frac{a_{k}(n)^{2}}{n^{1-2\kappa}}\right)^{\tfrac{1}{2}}\right).$

Similarly, we have

$\displaystyle S_{R,2}=$
$\displaystyle-2\left(\frac{1}{2\pi}\int_{1}^{T}\left(\log\frac{t}{2\pi}\right)dt\right)\sum_{n}\frac{\log*a_{k-1}(n)\cdot
a_{k}(n)}{n}$ $\displaystyle+\left(\left(\log
T+\int^{T}_{1}|(\log\frac{t}{2\pi})^{\prime}|dt\right)\left(\sum^{\infty}_{n=1}\frac{(\log*a_{k-1})(n)^{2}}{n^{2\kappa-1}}\right)^{\tfrac{1}{2}}\left(\sum^{\infty}_{n=1}\frac{a_{k}(n)^{2}}{n^{1-2\kappa}}\right)^{\tfrac{1}{2}}\right).$

Also,

$\displaystyle S_{R,3}=$
$\displaystyle\left(\frac{1}{2\pi}\int_{1}^{T}1dt\right)\sum_{n}\frac{(\Lambda*\log)*a_{k-1}(n)\cdot
a_{k}(n)}{n}+\left(\left(\sum^{\infty}_{n=1}\frac{(\Lambda*\log)*a_{k-1}(n)^{2}}{n^{2\kappa-1}}\right)^{\tfrac{1}{2}}\left(\sum^{\infty}_{n=1}\frac{a_{k}(n)^{2}}{n^{1-2\kappa}}\right)^{\tfrac{1}{2}}\right).$

We now apply the estimations given in (3.3) to see that for $T$ large enough,

$\displaystyle\sum^{\infty}_{n=1}\frac{a_{k}(n)^{2}}{n^{1-2\kappa}}\ll
e^{4k\log T/\log\log T}\sum_{n\leq
T^{40e^{2}k10^{-M/4}}}\frac{1}{n^{1-2\kappa}}\ll T^{1-\varepsilon}.$

Moreover, using the estimation $(\Lambda*\log)(n)\leq\log
n\sum_{d|n}\Lambda(d)=\log^{2}n$, we see that

(3.24) $\displaystyle\begin{split}&1*a_{k-1}(n)\leq\sum_{n\leq
T^{40e^{2}k10^{-M/4}}}a_{k-1}(n)\leq T^{1/2-\varepsilon},\\\
&(\log)*a_{k-1}(n)\leq\log n\sum_{n\leq T^{40e^{2}k10^{-M/4}}}a_{k-1}(n)\leq
T^{1/2-\varepsilon}\log n,\\\
&(\Lambda*\log)*a_{k-1}(n)\leq\log^{2}n\sum_{n\leq
T^{40e^{2}k10^{-M/4}}}a_{k-1}(n)\leq T^{1/2-\varepsilon}\log^{2}n.\end{split}$

It follows that

$\displaystyle\sum^{\infty}_{n=1}\frac{1*a_{k-1}(n)^{2}+(\log)*a_{k-1}(n)^{2}+(\Lambda*\log)*a_{k-1}(n)^{2}}{n^{2\kappa-1}}\ll
T^{1-2\varepsilon}\sum^{\infty}_{n=1}\frac{\log^{4}n}{n^{2\kappa-1}}\ll
T^{1-\varepsilon},$

where the last estimation above follows from the bound that (see [MN1, (16)])
uniformly for $\sigma>1$ and any integer $i\geq 0$,

(3.25)
$\displaystyle\sum^{\infty}_{n=1}\frac{\log^{i}n}{n^{\sigma}}\ll\frac{1}{(\sigma-1)^{i+1}}.$

We apply the above estimations in the evaluations of $S_{R,1},S_{R,2}$ and
$S_{R,3}$ to see that the contributions from the error terms can be ignored.
Furthermore, in the evaluation of $S_{R,1}$, we see that, for a monic
polynomial $\mathcal{Q}_{2}$ of degree $2$,

$\displaystyle\frac{1}{2\pi}\int_{1}^{T}\left(\log^{2}\frac{t}{2\pi}\right)dt=\frac{T}{2\pi}\mathcal{Q}_{2}(\mathcal{L})+O(1),$

where we denote $\mathcal{L}=\log(T/(2\pi))$.

Now, error term above contributes to an negligible error term since by (3.3)
and (3.24), we have

$\displaystyle\sum_{n}\frac{1*a_{k-1}(n)\cdot a_{k}(n)}{n}\ll
T^{1/2-\varepsilon}\sum_{n\leq T^{40e^{2}k10^{-M/4}}}\frac{1}{n}\ll
T^{1-\varepsilon}.$

We treat the integrals in the expressions for $S_{R,2}$ and $S_{R,3}$
similarly to arrive that, for monic polynomials $\mathcal{Q}_{i},i=1,2$ of
degree $i$,

(3.26) $\displaystyle S_{R}=$
$\displaystyle\frac{T}{2\pi}\sum_{n,m}\frac{a_{k-1}(m)a_{k}(mn)}{mn}\left(\mathcal{Q}_{2}(\mathcal{L})-2\mathcal{Q}_{1}(\mathcal{L})(\log
n)+((\Lambda*\log)(n)\right)+O(T^{1-\varepsilon}).$

Next, we evaluate $I_{L}$ using the notation given in (3.2) to see that

(3.27) $\displaystyle
I_{L}=\sum_{m}\sum_{n}\frac{a_{k-1}(m)a_{k}(n)}{m}\sum_{k}(\Lambda*\log)(k)\left(\frac{1}{2\pi
i}\int_{\kappa+i}^{\kappa+iT}\chi(1-s)\left(\frac{nk}{m}\right)^{-s}\right)\,ds\
.$

To evaluate the integral above, we need the following result from [MN1, Lemma
5.2].

######  Lemma 3.8.

Let $r,\kappa_{0}>0$. We have uniformly for $\kappa_{0}\leq\kappa\leq 2$ that

$\frac{1}{2\pi
i}\int_{\kappa+i}^{\kappa+iT}\chi(1-s)r^{-s}\,ds=I_{[0,T/2\pi]}(r)e(r)+O\left(r^{-\kappa}\left(T^{\kappa-1/2}+\frac{T^{\kappa+1/2}}{|T-2\pi
r|+T^{1/2}}\right)\right).$

where we write $e(x)=e^{2\pi ix}$ and we denote $I_{[0,T/2\pi]}(r)$ for the
indicator function of the interval $[0,\frac{T}{2\pi}]$, namely,
$I_{[0,T/2\pi]}(r)=1$ if $r\in[0,\frac{T}{2\pi}]$ and $I_{[0,T/2\pi]}(r)=0$
otherwise.

We apply Lemma 3.8 to see that the contribution from the error term in (3.27)
is

$\displaystyle\ll\sum_{m}\sum_{n}\frac{a_{k-1}(m)a_{k}(n)}{m}\sum_{k}(\Lambda*\log)(k)\left(\frac{nk}{m}\right)^{-\kappa}\left(T^{\kappa-1/2}+\frac{T^{\kappa+1/2}}{|T-2\pi\left(\frac{nk}{m}\right)|+T^{1/2}}\right).$

Using (3.25), we see that

$\displaystyle\sum^{\infty}_{k=1}\frac{(\Lambda*\log)(k)}{k^{\kappa}}\leq\sum^{\infty}_{k=1}\frac{\log^{2}(k)}{k^{\kappa}}\ll\frac{1}{(\kappa-1)^{3}}.$

We deduce from this and (3.3) that

$\displaystyle\ll\sum_{m}\sum_{n}\frac{a_{k-1}(m)a_{k}(n)}{m}\sum_{k}(\Lambda*\log)(k)\left(\frac{nk}{m}\right)^{-\kappa}T^{\kappa-1/2}=O(T^{1-\varepsilon}).$

We now use the ideas in the proof of [CGG, Lemma 2] to estimate

$\displaystyle\sum_{m}\sum_{n}\frac{a_{k-1}(m)a_{k}(n)}{m}\sum_{k}(\Lambda*\log)(k)\left(\frac{nk}{m}\right)^{-\kappa}\frac{T^{\kappa+1/2}}{|T-2\pi\left(\frac{nk}{m}\right)|+T^{1/2}}.$

We break up the sum into three parts. The terms with
$|T-2\pi\left(\frac{nk}{m}\right)|>\tfrac{1}{2}T$ contribute
$O(T^{1-\varepsilon})$ as our discussions above. The terms
$T^{\tfrac{1}{2}}\leq|T-2\pi\left(\frac{nk}{m}\right)|\leq\tfrac{1}{2}T$ are
further divided into cases that $T^{\tfrac{1}{2}}\leq
T-2\pi\left(\frac{nk}{m}\right)\leq\tfrac{1}{2}T$ and $T^{\tfrac{1}{2}}\leq
2\pi\left(\frac{nk}{m}\right)-T\leq\tfrac{1}{2}T$. Without loss of generality,
we consider the case that $T^{\tfrac{1}{2}}\leq
2\pi\left(\frac{nk}{m}\right)-T\leq\tfrac{1}{2}T$. This implies that
$T+T^{\tfrac{1}{2}}\leq 2\pi\left(\frac{nk}{m}\right)\leq T+\tfrac{1}{2}T$. We
then split the sum into $\ll\log T$ sums of the shape

$\displaystyle T+P\leq 2\pi\left(\frac{nk}{m}\right)\leq T+2P$

where $T^{\tfrac{1}{2}}\ll P\ll T$. The above implies that

(3.28) $\displaystyle(nk)^{-\kappa}\ll(mT)^{-\kappa}.$

Moreover, the sum over $n,k$ ranges over an interval of length $\ll mP$, Thus,
the contribution form the corresponding terms (using $a_{k}(n)\ll
T^{\varepsilon}$, $(\Lambda*\log)(k)\ll\log^{2}k\ll\log^{2}(m(T+P))\ll
T^{\varepsilon}$, $1/(|T-2\pi\left(\frac{nk}{m}\right)|+T^{1/2})\ll 1/P$) is

$\displaystyle
T^{\varepsilon}\sum_{m}\sum_{n}\frac{a_{k-1}(m)}{m}mP(mT)^{-\kappa}\frac{T^{\kappa+1/2}}{P}\ll
T^{1-\varepsilon}.$

Lastly, we consider the contributions from the terms
$|T-2\pi\left(\frac{nk}{m}\right)|\leq T^{\tfrac{1}{2}}$ by noticing that in
this case the estimation (3.28) is still valid. Moreover, the sum over $n,k$
ranges over an interval of length $\ll mT^{\tfrac{1}{2}}$, Thus, the
contribution form the corresponding terms is

$\displaystyle\ll
T^{\varepsilon}\sum_{m}\sum_{n}\frac{a_{k-1}(m)}{m}mT^{\tfrac{1}{2}}(mT)^{-\kappa}\frac{T^{\kappa+1/2}}{T^{\tfrac{1}{2}}}\ll
T^{1-\varepsilon}.$

We then conclude from the above discussions that

$\displaystyle I_{L}=\sum_{m}\sum_{n}\frac{a_{k-1}(m)a_{k}(n)}{m}\sum_{1\leq
k\leq\frac{mT}{2\pi n}}e(-\frac{kn}{m})+O(T^{1-\varepsilon}).$

We now proceed as in [MN1, p. 3212-3213] to see that, for a monic polynomial
$\mathcal{P}_{2}$ of degree $2$,

(3.29)
$\displaystyle\begin{split}I_{L}=&\frac{T}{4\pi}\sum_{n,m}\frac{a_{k-1}(m)a_{k}(mn)}{mn}\mathcal{P}_{2}(\log\frac{T}{2\pi
n})+O(T^{1-\varepsilon})+O(\sum_{n,m}\frac{a_{k-1}(m)a_{k}(n)}{mn})(m,n)\Big{(}\Lambda_{2}(\frac{m}{(m,n)})+\Lambda(\frac{m}{(m,n)})\log
T\Big{)}.\end{split}$

To estimate the last error term in (3.29), we write $d=(m,n),m=dL,n=dN$ and
use the easily checked property that $a_{k-1}(mn)\leq a_{k-1}(m)a_{k-1}(n)$ to
see that it is

$\displaystyle\ll\sum_{d,N}\frac{a_{k-1}(d)a_{k}(dN)}{dN}\sum_{L}\frac{a_{k-1}(L)}{L}\Big{(}\Lambda_{2}(L)+\Lambda(L)\log
T\Big{)}.$

Using (2.4) and the observation from (3.3) that $a_{k-1}(L)$ is bounded when
$L$ is supported on integers $L$ with $\omega(L)\leq 2$ and $a_{k-1}(L)\neq 0$
only for $L\leq T^{40e^{2}k10^{-M/4}}$, we proceed as in [MN1, p. 3213] to see
that

$\displaystyle\sum_{L}\frac{a_{k-1}(L)}{L}\Big{(}\Lambda_{2}(L)+\Lambda(L)\log
T\Big{)}\ll 10^{-M/4}\log^{2}T.$

We conclude from (3.26), (3.29) and the above estimation that

$\displaystyle S\geq$
$\displaystyle\frac{T}{2\pi}\sum_{n,m}\frac{a_{k-1}(m)a_{k}(mn)}{mn}\left(\mathcal{Q}_{2}(\mathcal{L})-2\mathcal{Q}_{1}(\mathcal{L})(\log
n)-\tfrac{1}{2}\mathcal{P}_{2}(\mathcal{L}-\log n)+((\Lambda*\log)(n)\right)$
$\displaystyle+O(10^{-M/4}T\log^{2}T\sum_{n,m}\frac{a_{k-1}(m)a_{k}(mn)}{mn})+O(T^{1-\varepsilon}).$

We now take $M$ large enough and argue as in [MN1, p. 3214] to see that

$\displaystyle S\gg$ $\displaystyle
T\log^{2}T\sum_{n,m}\frac{a_{k-1}(m)a_{k}(mn)}{mn}.$

Lastly, we apply arguments used in [Gao2021-4, Section 4] to estimate the sums
above to arrive that

$\displaystyle S\gg$ $\displaystyle T(\log T)^{k^{2}+2}.$

This completes the proof of the proposition.

###  3.9. Proof of Proposition 3.5

Upon dividing the range of $\gamma$ into dyadic blocks and replacing $T$ by
$2T$, we see that it suffices to show for large $T$,

(3.30) $\displaystyle\sum_{T<\gamma\leq
2T}\prod^{\mathcal{J}}_{j=1}\big{(}|{\mathcal{N}}_{j}(\rho,k)|^{2}+|{\mathcal{Q}}_{j}(\rho,k)|^{2r_{k}}\big{)}\ll$
$\displaystyle T(\log T)^{k^{2}+1}.$

We apply Lemma 2.1 to evaluate the left side expression above. In this
process, we may ignore the contributions from the error terms in (2.2), using
arguments similar to our treatments on various error terms in the proof of
Proposition 3.4. Moreover, we may also ignore the contributions from the main
terms in (2.2) from the cases $a\neq b$, since these terms are negative and we
are seeking for an upper bound here. Thus, only the diagonal terms in the left
side expression of (3.30) survive. Now applying (2.3), we conclude that

$\displaystyle\begin{split}&\sum_{T<\gamma\leq
2T}\prod^{\mathcal{J}}_{j=1}\big{(}|{\mathcal{N}}_{j}(\rho,k)|^{2}+|{\mathcal{Q}}_{j}(\rho,k)|^{2r_{k}}\big{)}\\\
\ll&T(\log
T)\prod^{\mathcal{J}}_{j=1}\Big{(}\sum_{n_{j}}\frac{k^{2\Omega(n_{j})}}{n_{j}g^{2}(n_{j})}b_{j}(n_{j})+\Big{(}\frac{64\max(2,k+3/2)}{\lceil
e^{2}k\alpha^{-3/4}_{j}\rceil}\Big{)}^{2r_{k}\lceil
e^{2}k\alpha^{-3/4}_{j}\rceil}((r_{k}\lceil
e^{2}k\alpha^{-3/4}_{j}\rceil)!)^{2}\sum_{\begin{subarray}{c}\Omega(n_{j})=r_{k}\lceil
e^{2}k\alpha^{-3/4}_{j}\rceil\\\ p|n_{j}\implies p\in
I_{j}\end{subarray}}\frac{1}{n_{j}g^{2}(n_{j})}\Big{)}.\end{split}$

The product above is analogues to the product in [Gao2021-4, (6.20)]. Using
similar estimations, we see that it is $\ll(\log T)^{k^{2}}$. This implies
(3.30) and completes the proof of the proposition.

###  3.10. Proof of Proposition 3.6

Again by dividing the range of $\gamma$ into dyadic blocks and replacing $T$
by $2T$, we see that it suffices to show for large $T$,

(3.31) $\displaystyle\sum_{T<\gamma\leq
2T}|\zeta^{\prime}(\rho)|^{2}|\mathcal{N}(\rho,k-1)|^{2}\ll T(\log
T)^{k^{2}+3}.$

We split the interval $(0,T^{\alpha_{\mathcal{J}}}]$ into disjoint
subintervals $I_{j}=(T^{\alpha_{j-1}},T^{\alpha_{j}}]$ for $1\leq
j\leq\mathcal{J}$ and define

$w_{j}(n)=\frac{\Lambda_{\mathcal{L}}(n)}{n^{1/(\alpha_{j}\log T)}\log
n}\frac{\log(T^{\alpha_{j}}/n)}{\log T^{\alpha_{j}}},$

where

$\displaystyle\Lambda_{\mathcal{L}}(n)=\begin{cases}\Lambda(n),\quad\text{ if
}n=p\text{ or if }n=p^{2}\text{ and }n\leq\mathcal{L},\\\
0,\quad\text{otherwise}.\end{cases}$

For $1\leq l\leq j\leq\mathcal{J}$, we define

$G_{l,j}(t)=\Re\sum_{n\in I_{l}}\frac{w_{j}(n)}{\sqrt{n}}n^{-it}.$

Note that our definition of $G_{l,j}$ is slightly different from that in
[Kirila], due to our definition on $\mathcal{L}$. However, we notice the
bounds

(3.32) $\displaystyle\begin{split}w_{j}(p)\leq 1,\quad
w_{j}(p^{2})\leq\tfrac{1}{2},\end{split}$

so that it follows from the above and Lemma 2.2 that

$\sum_{\mathcal{L}\leq p\leq\log T}\frac{w_{j}(p^{2})}{p}=O(1).$

We then deduce from [Kirila, (4.1)] the following upper bound for
$\log|\zeta^{\prime}(\rho)|$, which says that for any $1\leq
j\leq\mathcal{J}$,

(3.33)
$\displaystyle\begin{split}&\log|\zeta^{\prime}(\rho)|\ll\sum^{j}_{l=1}G_{l,j}(\gamma)+\log\log
T+\alpha^{-1}_{j}+O(1).\end{split}$

We also define the following sets:

$\displaystyle\mathcal{S}(0)=$ $\displaystyle\\{T<\gamma\leq
2T:|G_{1,l}(\gamma)|>\alpha_{1}^{-3/4}\;\text{ for some }1\leq
l\leq\mathcal{J}\\},$ $\displaystyle\mathcal{S}(j)=$
$\displaystyle\\{T<\gamma\leq
2T:|G_{m,l}(\gamma)|\leq\alpha_{m}^{-3/4}\;\forall 1\leq m\leq j,\;\forall
m\leq l\leq\mathcal{J},$ $\displaystyle\;\;\;\;\;\text{but
}|G_{j+1,l}(\gamma)|>\alpha_{j+1}^{-3/4}\;\text{ for some }j+1\leq
l\leq\mathcal{J}\\},\quad 1\leq j\leq\mathcal{J},$
$\displaystyle\mathcal{S}(\mathcal{J})=$ $\displaystyle\\{T<\gamma\leq
2T:|G_{m,\mathcal{J}}(\gamma)|\leq\alpha_{m}^{-3/4}\;\forall 1\leq
m\leq\mathcal{J}\\},$

so that

$\\{T<\gamma\leq 2T\\}=\bigcup_{j=0}^{\mathcal{J}}\mathcal{S}(j).$

It follows that

(3.34) $\displaystyle\sum_{T<\gamma\leq
2T}|\zeta^{\prime}(\rho)|^{2}|\mathcal{N}(\rho,k-1)|^{2}=\sum_{j=0}^{\mathcal{J}}\sum_{\gamma\in
S(j)}|\zeta^{\prime}(\rho)|^{2}|\mathcal{N}(\rho,k-1)|^{2}.$

We note from [Kirila, Lemma 5.5] and (2.3) that we have

(3.35) $\displaystyle\text{meas}(\mathcal{S}(0))\ll$ $\displaystyle T(\log
T)e^{-(\log\log T)^{2}/10}.$

We then deduce via the Cauchy-Schwarz inequality that

(3.36) $\displaystyle\begin{split}&\sum_{\gamma\in
S(0)}|\zeta^{\prime}(\rho)|^{2}|\mathcal{N}(\rho,k-1)|^{2}\leq\Big{(}\text{meas}(\mathcal{S}(0))\Big{)}^{1/4}\Big{(}\sum_{T<\gamma\leq
2T}|\zeta^{\prime}(\rho)|^{8}\Big{)}^{1/4}\Big{(}\sum_{T<\gamma\leq
2T}|\mathcal{N}(\rho,k-1)|^{4}\Big{)}^{1/2}.\end{split}$

Similar to the proof of Proposition 3.5, we have that

(3.37) $\displaystyle\sum_{T<\gamma\leq 2T}|\mathcal{N}(\rho,k-1)|^{4}\ll
T(\log T)^{(2k-2))^{2}+1}.$

Also, applying (1.3) with $k=4,\varepsilon=1$, we see that

(3.38) $\displaystyle\sum_{T<\gamma\leq 2T}|\zeta^{\prime}(\rho)|^{8}\leq
T(\log T)^{25}.$

We use the bounds given in (3.35), (3.37) and (3.38) in (3.36) to conclude
that

(3.39)
$\displaystyle\sum_{\gamma\in\mathcal{S}(0)}|\zeta^{\prime}(\rho)|^{2}|\mathcal{N}(\rho,k-1)|^{2}\ll
T(\log T)^{k^{2}+3}.$

The above estimation implies that it remains to consider the cases $j\geq 1$
in (3.34). Without loss of generality, we may assume that $1\leq
j\leq\mathcal{J}-1$ here. When $\gamma\in\mathcal{S}(j)$, we deduce from
(3.33) that

$\displaystyle\begin{split}&\sum_{\gamma\in\mathcal{S}(j)}|\zeta^{\prime}(\rho)|^{2}|\mathcal{N}(\rho,k-1)|^{2}\ll(\log
T)^{2}\exp\big{(}\frac{2}{\alpha_{j}}\big{)}\sum_{\gamma\in\mathcal{S}(j)}\exp\Big{(}2\sum^{j}_{l=1}G_{l,j}(\gamma)\Big{)}|\mathcal{N}(\rho,k-1)|^{2}.\end{split}$

As we have $G_{l,j}\leq\alpha^{-3/4}_{l}$ when $\gamma\in\mathcal{S}(j)$, we
apply [Kirila, Lemma 5.2] to see that

$\displaystyle\begin{split}\exp\Big{(}2\sum^{j}_{l=1}G_{l,j}(\gamma)\Big{)}\ll\prod^{j}_{l=1}E^{2}_{e^{2}k\alpha^{-3/4}_{l}}(G_{l,j}(\gamma)).\end{split}$

We then deduce from the description on $\mathcal{S}(j)$ that when $j\geq 1$,

(3.40)
$\displaystyle\begin{split}&\sum_{\gamma\in\mathcal{S}(j)}|\zeta^{\prime}(\rho)|^{2}|\mathcal{N}(\rho,k-1)|^{2}\\\
\ll&(\log
T)^{2}\exp\big{(}\frac{2}{\alpha_{j}}\big{)}\sum^{\mathcal{I}}_{m=j+1}\sum_{\gamma\in\mathcal{S}(j)}\exp\Big{(}2\sum^{j}_{l=1}G_{l,j}(\gamma)\Big{)}|\mathcal{N}(\rho,k-1)|^{2}\Big{(}\alpha^{3/4}_{j+1}G_{j+1,m}(\gamma)\Big{)}^{2\lceil
1/(10\alpha_{j+1})\rceil}\\\ \ll&(\log
T)^{2}\exp\big{(}\frac{2}{\alpha_{j}}\big{)}\sum^{\mathcal{I}}_{m=j+1}\sum_{\gamma\in\mathcal{S}(j)}\prod^{j}_{l=1}E^{2}_{e^{2}k\alpha^{-3/4}_{l}}(G_{l,j}(\gamma))|E_{e^{2}k\alpha^{-3/4}_{l}}((k-1){\mathcal{P}}_{l}(\rho)|^{2}\\\
&\times|E_{e^{2}k\alpha^{-3/4}_{j+1}}((k-1){\mathcal{P}}_{j+1}(\rho)|^{2}\Big{(}\alpha^{3/4}_{j+1}G_{j+1,m}(\gamma)\Big{)}^{2\lceil
1/(10\alpha_{j+1})\rceil}\prod^{\mathcal{J}}_{n=j+2}|E_{e^{2}k\alpha^{-3/4}_{n}}((2k-2){\mathcal{P}}_{n}(\rho))|^{2}.\end{split}$

Note that

(3.41)
$\displaystyle\begin{split}&\sum_{\gamma\in\mathcal{S}(j)}\prod^{j}_{l=1}E^{2}_{e^{2}k\alpha^{-3/4}_{l}}(G_{l,j}(\gamma))|E_{e^{2}k\alpha^{-3/4}_{l}}((k-1){\mathcal{P}}_{l}(\rho)|^{2}\\\
&\times|E_{e^{2}k\alpha^{-3/4}_{j+1}}((k-1){\mathcal{P}}_{j+1}(\rho)|^{2}\Big{(}\alpha^{3/4}_{j+1}G_{j+1,m}(\gamma)\Big{)}^{2\lceil
1/(10\alpha_{j+1})\rceil}\prod^{\mathcal{J}}_{n=j+2}|E_{e^{2}k\alpha^{-3/4}_{n}}((2k-2){\mathcal{P}}_{n}(\rho))|^{2}\\\
\leq&\sum_{T<\gamma\leq
2T}\prod^{j}_{l=1}E^{2}_{e^{2}k\alpha^{-3/4}_{l}}(G_{l,j}(\gamma))|E_{e^{2}k\alpha^{-3/4}_{l}}((k-1){\mathcal{P}}_{l}(\rho)|^{2}\\\
&\times|E_{e^{2}k\alpha^{-3/4}_{j+1}}((k-1){\mathcal{P}}_{j+1}(\rho)|^{2}\Big{(}\alpha^{3/4}_{j+1}G_{j+1,m}(\gamma)\Big{)}^{2\lceil
1/(10\alpha_{j+1})\rceil}\prod^{\mathcal{J}}_{n=j+2}|E_{e^{2}k\alpha^{-3/4}_{n}}((2k-2){\mathcal{P}}_{n}(\rho))|^{2}.\end{split}$

We shall apply Lemma 2.1 to evaluate the last sum above. As in the case for
the proof of Proposition 3.4, we may only focus on the main term in the
process. To facilitate our evaluation of the last sum above, we follow the
treatments in [Kirila] by introducing a sequence of independent random
variables $\\{X_{p}\\}$ such that each $X_{p}$ is uniformly distributed on the
unit circle in the complex plane. We also define

$X_{n}=X_{p_{1}}^{h_{1}}\cdots X_{p_{r}}^{h_{r}}$

for $n=p_{1}^{h_{1}}\cdots p_{r}^{h_{r}}$ so that $X_{n}$ is a random
completely multiplicative function. We then define random models $G_{l,j}(X)$
for $1\leq l\leq j\leq{\mathcal{J}}$ by

$G_{l,j}(X)=\Re\sum_{n\in
I_{l}}\frac{w_{j}(n)}{\sqrt{n}}X_{n},\quad{\mathcal{P}}_{j}(X)=\sum_{p\in
I_{j}}\frac{1}{\sqrt{p}}X_{p}.$

Similar to [Kirila, Lemma 5.3], we have under RH and other than a negligible
error term, for $1\leq l\leq j\leq{\mathcal{J}}$ and non-negative integers
$n_{l,1},n_{l,2},n_{l,3}$ satisfying
$\max(n_{l,1}/2,n_{l,2},n_{l,3})\leq\max(\lceil
e^{2}k\alpha^{-3/4}_{l}\rceil,2\lceil 1/(10\alpha_{j+1})\rceil)$,

$\displaystyle\begin{split}&\sum_{T<\gamma\leq
2T}\prod^{{\mathcal{J}}}_{l=1}G^{n_{l,1}}_{l,j}(\gamma){\mathcal{P}}^{n_{l,2}}_{l}(\rho){\mathcal{P}}^{n_{l,3}}_{l}(\overline{\rho})\leq
N(T,2T)\mathbb{E}\Big{(}\prod^{{\mathcal{J}}}_{l=1}G^{n_{l,1}}_{l,j}(X){\mathcal{P}}^{n_{l,2}}_{l}(X){\mathcal{P}}^{n_{l,3}}_{l}(\overline{X})\Big{)}.\end{split}$

We now proceed as in the proof of [Kirila, Lemma 5.5] to see that, under RH
and other than a negligible error term, the last sum in (3.41) is

$\displaystyle\begin{split}\ll&\Big{(}\alpha^{3/4}_{j+1}\Big{)}^{2\lceil
1/(10\alpha_{j+1})\rceil}N(T,2T)\mathbb{E}\Big{(}G_{j+1,m}(X)^{2\lceil
1/(10\alpha_{j+1})\rceil}\exp\Big{(}2\sum^{j}_{l=1}G_{l,j}(X)+2(k-1)\sum^{{\mathcal{J}}}_{n=1}\Re{\mathcal{P}}_{n}(X)\Big{)}\Big{)}\\\
=&\Big{(}\alpha^{3/4}_{j+1}\Big{)}^{2\lceil
1/(10\alpha_{j+1})\rceil}N(T,2T)\mathbb{E}\Big{(}\exp\Big{(}2G_{1,j}(X)+2(k-1)\Re{\mathcal{P}}_{1}(X)\Big{)}\Big{)}\prod^{j}_{l=2}\mathbb{E}\Big{(}\exp\Big{(}2G_{l,j}(X)+2(k-1)\Re{\mathcal{P}}_{l}(X)\Big{)}\Big{)}\\\
&\times\mathbb{E}\Big{(}G_{j+1,m}(X)^{2\lceil
1/(10\alpha_{j+1})\rceil}\exp\Big{(}2(k-1)\Re{\mathcal{P}}_{j+1}(X)\Big{)}\Big{)}\times\prod^{{\mathcal{J}}}_{l=j+2}\mathbb{E}\Big{(}\exp\Big{(}2(k-1)\Re{\mathcal{P}}_{l}(X)\Big{)}.\end{split}$

Similar to the evaluation done on [Kirila, p. 492], we see that

(3.42)
$\displaystyle\begin{split}\prod^{j}_{l=2}\mathbb{E}\Big{(}\exp\Big{(}2G_{l,j}(X)+2(k-1)\Re{\mathcal{P}}_{l}(X)\Big{)}\Big{)}=&\prod^{j}_{l=2}\prod_{p\in
I_{l}}I_{0}\big{(}\frac{2w_{j}(p)}{\sqrt{p}}+\frac{2(k-1)}{\sqrt{p}}\big{)},\\\
\prod^{{\mathcal{J}}}_{l=j+2}\mathbb{E}\Big{(}\exp\Big{(}2(k-1)\Re{\mathcal{P}}_{l}(X)\Big{)}\Big{)}=&\prod^{{\mathcal{J}}}_{l=j+2}\prod_{p\in
I_{l}}I_{0}\big{(}2(k-1)\frac{1}{\sqrt{p}}\big{)},\end{split}$

where $I_{0}(z)=\sum_{n=0}^{\infty}\frac{(z/2)^{2n}}{(n!)^{2}}$ is the
modified Bessel function of the first kind.

Next, using the arguments similar to those in [Kirila, (6.3)], we have that

$\displaystyle\begin{split}&\mathbb{E}\Big{(}\exp\Big{(}2G_{1,j}(X)+2(k-1)\Re{\mathcal{P}}_{1}(X)\Big{)}\Big{)}\\\
=&\prod_{\begin{subarray}{c}p\in I_{1}\\\ p>\log
T\end{subarray}}I_{0}\big{(}\frac{2w_{j}(p)}{\sqrt{p}}+\frac{2(k-1)}{\sqrt{p}}\big{)}\\\
&\times\mathbb{E}\Big{(}\exp\Big{(}\sum_{p\leq\log
T}\big{(}\frac{2w_{j}(p)}{\sqrt{p}}+\frac{2(k-1)}{\sqrt{p}}\big{)}\Re
X_{p}+\sum_{p\leq\log T}\frac{4w_{j}(p^{2})}{\sqrt{p}}(\Re
X_{p}^{2})-\sum_{p\leq\log
T}\frac{4w_{j}(p^{2})}{\sqrt{p}}\Big{)}\Big{)}.\end{split}$

We apply the bounds given in (3.32) and proceed as in [Kirila, p. 492-493] to
deduce that the last expression above is

$\displaystyle\begin{split}\ll\prod_{\begin{subarray}{c}p\in
I_{1}\end{subarray}}I_{0}\big{(}\frac{2w_{j}(p)}{\sqrt{p}}+\frac{2(k-1)}{\sqrt{p}}\big{)}.\end{split}$

Furthermore, we notice that

(3.43) $\displaystyle\begin{split}&\mathbb{E}\Big{(}G_{j+1,m}(X)^{2\lceil
1/(10\alpha_{j+1})\rceil}\exp\Big{(}2(k-1)\Re{\mathcal{P}}_{j+1}(X)\Big{)}\Big{)}\\\
=&\sum^{\infty}_{n=0}\mathbb{E}\Big{(}G_{j+1,m}(X)^{2\lceil
1/(10\alpha_{j+1})\rceil}\frac{(2(k-1)\Re{\mathcal{P}}_{j+1}(X))^{n}}{n!}\Big{)}\\\
=&\sum^{\infty}_{n=0}\mathbb{E}\Big{(}G_{j+1,m}(X)^{2\lceil
1/(10\alpha_{j+1})\rceil}\frac{(2(k-1)\Re{\mathcal{P}}_{j+1}(X))^{2n}}{(2n)!}\Big{)}\\\
\leq&\sum^{\infty}_{n=0}\mathbb{E}\Big{(}\frac{(2\Re{\mathcal{P}}_{j+1}(X))^{2\lceil
1/(10\alpha_{j+1})\rceil+2n}}{(2n)!}\Big{)},\end{split}$

where the last estimation above follows from the observation that, upon using
(3.32) and noting that $0<k\leq 1/2$, we have for any integer $n\geq 0$,

(3.44) $\displaystyle\begin{split}&\mathbb{E}\Big{(}G_{j+1,m}(X)^{2\lceil
1/(10\alpha_{j+1})\rceil}\frac{(2(k-1)\Re{\mathcal{P}}_{j+1}(X))^{2n}}{(2n)!}\Big{)}\leq\
\mathbb{E}\Big{(}\frac{(2\Re{\mathcal{P}}_{j+1}(X))^{2\lceil
1/(10\alpha_{j+1})\rceil+2n}}{(2n)!}\Big{)}.\end{split}$

Observe that for any positive integer $m$,

$\mathbb{E}[(\Re
X_{p})^{m}]=\begin{cases}\displaystyle\binom{2h}{h}2^{-m}&\hbox{if }m=2h,\\\
\displaystyle 0&\hbox{if }m\text{ is odd}.\end{cases}$

It follows from this that if we denote $\\{p|p\in
I_{j+1}\\}=\\{p_{1},\cdots,p_{s}\\}$, then we have for any positive integer
$m$,

$\displaystyle\begin{split}\mathbb{E}\Big{(}(2\Re{\mathcal{P}}_{j+1}(X))^{2m}\Big{)}=&\sum_{\begin{subarray}{c}m_{1},m_{2}\cdots,m_{s}\geq
0\\\
m_{1}+m_{2}+\cdots+m_{s}=m\end{subarray}}\binom{2m}{2m_{1},\cdots,2m_{s}}\prod^{s}_{i=1}(\frac{2}{\sqrt{p_{i}}})^{2m_{i}}\mathbb{E}(\Re
X_{p})^{2m_{i}}\\\
=&2^{-2m}\sum_{\begin{subarray}{c}m_{1},m_{2}\cdots,m_{s}\geq 0\\\
m_{1}+m_{2}+\cdots+m_{s}=m\end{subarray}}\binom{2m}{2m_{1},\cdots,2m_{s}}\prod^{s}_{i=1}\binom{2m_{i}}{m_{i}}(\frac{4}{p_{i}})^{m_{i}}\\\
\leq&\frac{(2m)!}{2^{2m}m!}(\sum_{p\in I_{j+1}}\frac{4}{p})^{m}.\end{split}$

We apply the above estimation to the last expression in (3.44) to deduce that

(3.45) $\displaystyle\begin{split}&\mathbb{E}\Big{(}G_{j+1,m}(X)^{2\lceil
1/(10\alpha_{j+1})\rceil}\frac{(2(k-1)\Re{\mathcal{P}}_{j+1}(X))^{2n}}{(2n)!}\Big{)}\\\
\leq&\ \frac{(2\lceil 1/(10\alpha_{j+1})\rceil+2n)!}{2^{2\lceil
1/(10\alpha_{j+1})\rceil+2n}(\lceil
1/(10\alpha_{j+1})\rceil+n)!(2n)!}\bigg{(}\sum_{p\in
I_{j+1}}\frac{4}{p}\bigg{)}^{\lceil 1/10\alpha_{j+1}\rceil+n}.\end{split}$

Using the bounds

(3.46) $\displaystyle(\frac{n}{e})^{n}\leq n!\leq n(\frac{n}{e})^{n},$

we see that the last expression in (3.45) is

(3.47) $\displaystyle\begin{split}\ll\frac{(2\lceil
1/(10\alpha_{j+1})\rceil+2n)}{(2n)!}\bigg{(}\frac{\lceil
1/(10\alpha_{j+1})\rceil+n}{e}\sum_{T^{\alpha_{j}}<p\leq
T^{\alpha_{j+1}}}\frac{4}{p}\bigg{)}^{\lceil
1/(10\alpha_{j+1})\rceil+n}.\end{split}$

Notice that when $n\geq 2\lceil 1/(10\alpha_{j+1})\rceil$, the expression
above is

$\displaystyle\begin{split}\leq\frac{3n}{(2n)!}\bigg{(}\frac{3n}{2e}\sum_{T^{\alpha_{j}}<p\leq
T^{\alpha_{j+1}}}\frac{4}{p}\bigg{)}^{3n/2}\leq
3n(\frac{e}{2n})^{2n}\bigg{(}\frac{60n}{e}\bigg{)}^{3n/2},\end{split}$

where the last estimation above follows from (3.1) and (3.46). As the sum of
the last term above over $n$ is convergent, we see that the contribution of
these terms to the last sum of (3.43) is $O(1)$ and may be ignored.

Now for $\frac{1}{4}\lceil 1/(10\alpha_{j+1})\rceil\leq n<2\lceil
1/(10\alpha_{j+1})\rceil$, we apply (3.1) to see that the expression in (3.47)
is

$\displaystyle\begin{split}\leq&\frac{6\lceil
1/(10\alpha_{j+1})\rceil}{(2n)!}\bigg{(}\frac{3\lceil
1/(10\alpha_{j+1})\rceil}{e}\sum_{T^{\alpha_{j}}<p\leq
T^{\alpha_{j+1}}}\frac{4}{p}\bigg{)}^{\lceil
1/(10\alpha_{j+1})\rceil}\cdot\bigg{(}\frac{5n}{e}\sum_{T^{\alpha_{j}}<p\leq
T^{\alpha_{j+1}}}\frac{4}{p}\bigg{)}^{n}\\\ \leq&\frac{6\lceil
1/(10\alpha_{j+1})\rceil}{(2n)!}\bigg{(}\frac{120\lceil
1/(10\alpha_{j+1})\rceil}{e}\bigg{)}^{\lceil
1/(10\alpha_{j+1})\rceil}\cdot\bigg{(}\frac{200n}{e}\bigg{)}^{n}.\end{split}$

Applying (3.46) again, we see that the sum of the last term above over $n$ is
convergent so that the contribution of these terms to the last sum of (3.43)
is

(3.48) $\displaystyle\begin{split}\ll\lceil
1/(10\alpha_{j+1})\rceil\bigg{(}\frac{120\lceil
1/(10\alpha_{j+1})\rceil}{e}\bigg{)}^{\lceil
1/(10\alpha_{j+1})\rceil}.\end{split}$

Lastly, for $n<\frac{1}{4}\lceil 1/(10\alpha_{j+1})\rceil$, we apply (3.1) and
(3.46) one more time to see that the expression in (3.47) is

$\displaystyle\begin{split}\leq&\frac{5\lceil
1/(10\alpha_{j+1})\rceil/2}{(2n)!}\bigg{(}\frac{5\lceil
1/(10\alpha_{j+1})\rceil}{4e}\sum_{T^{\alpha_{j}}<p\leq
T^{\alpha_{j+1}}}\frac{4}{p}\bigg{)}^{5\lceil 1/(10\alpha_{j+1})\rceil/4}\\\
\leq&\frac{5\lceil 1/(10\alpha_{j+1})\rceil/2}{(2n)!}\bigg{(}\frac{200\lceil
1/(10\alpha_{j+1})\rceil}{4e}\bigg{)}^{5\lceil
1/(10\alpha_{j+1})\rceil/4}.\end{split}$

Upon summing over $n$, we see that the contribution of these terms to the last
sum of (3.43) is

(3.49) $\displaystyle\begin{split}\ll\lceil
1/(10\alpha_{j+1})\rceil\bigg{(}\frac{200\lceil
1/(10\alpha_{j+1})\rceil}{4e}\bigg{)}^{5\lceil
1/(10\alpha_{j+1})\rceil/4}.\end{split}$

We apply the bounds (3.48) and (3.49) in (3.43) to conclude that

$\displaystyle\begin{split}&\mathbb{E}\Big{(}G_{j+1,m}(X)^{2\lceil
1/(10\alpha_{j+1})\rceil}\exp\Big{(}2(k-1)\Re{\mathcal{P}}_{j+1}(X)\Big{)}\Big{)}\ll\lceil
1/(10\alpha_{j+1})\rceil\bigg{(}\frac{120\lceil
1/(10\alpha_{j+1})\rceil}{e}\bigg{)}^{5\lceil
1/(10\alpha_{j+1})\rceil/4}.\end{split}$

We combine (3.42), (3.43) and the last estimation above to see that, via using
(3.32), the last expression in (3.41) is

$\displaystyle\begin{split}\ll&\Big{(}\alpha^{3/4}_{j+1}\Big{)}^{2\lceil
1/(10\alpha_{j+1})\rceil}\lceil
1/(10\alpha_{j+1})\rceil\bigg{(}\frac{120\lceil
1/(10\alpha_{j+1})\rceil}{e}\bigg{)}^{5\lceil
1/(10\alpha_{j+1})\rceil/4}N(T,2T)\prod^{{\mathcal{J}}}_{l=1}\prod_{p\in
I_{l}}I_{0}\big{(}\frac{2w_{j}(p)}{\sqrt{p}}+\frac{2(k-1)}{\sqrt{p}}\big{)}\\\
\ll&\Big{(}\frac{1}{\alpha_{j+1}}\Big{)}^{-\lceil
1/(10\alpha_{j+1})\rceil/8}N(T,2T)\prod^{{\mathcal{J}}}_{l=1}\prod_{p\in
I_{l}}I_{0}\big{(}\frac{2k}{\sqrt{p}}\big{)}.\end{split}$

Using further the estimation that $I_{0}(2x)\leq e^{x^{2}}$, we deduce from
the above that the last expression in (3.41) is

(3.50)
$\displaystyle\begin{split}\ll\Big{(}\frac{1}{\alpha_{j+1}}\Big{)}^{-\lceil
1/(10\alpha_{j+1})\rceil/8}N(T,2T)\exp\Big{(}\sum_{p\leq
T^{\alpha_{{\mathcal{J}}}}}\frac{k^{2}}{p}\Big{)}\ll\Big{(}\frac{1}{\alpha_{j+1}}\Big{)}^{-\lceil
1/(10\alpha_{j+1})\rceil/8}T(\log T)^{k^{2}+1},\end{split}$

where the last estimation above follows from (3.50) and (2.3).

As $20/\alpha_{j+1}=1/\alpha_{j}$, we conclude from (3.34), (3.39), (3.40),
(3.41) and (3.50) and that

$\displaystyle\begin{split}\sum_{T<\gamma\leq
2T}|\zeta^{\prime}(\rho)|^{2}|\mathcal{N}(\rho,k-1)|^{2}\ll&\sum_{j=1}^{\mathcal{J}}\sum_{\gamma\in
S(j)}|\zeta^{\prime}(\rho)|^{2}|\mathcal{N}(\rho,k-1)|^{2}+T(\log
T)^{k^{2}+3}\\\ \ll&T(\log
T)^{k^{2}+3}\sum_{j=1}^{\mathcal{J}}\Big{(}\frac{1}{\alpha_{j+1}}\Big{)}^{-\lceil
1/(10\alpha_{j+1})\rceil/(10)}.\end{split}$

As the sum of the right side expression over $j$ converges, we see that the
above estimation implies (3.31) and this completes the proof of Proposition
3.6.

Acknowledgments. P. G. is supported in part by NSFC grant 11871082.

## References


A commutative ring $R$ is a tca concentrated in degree $0$, where the
symmetric group acts trivially.

######  Example 4.7.

For $m\leq\frac{w}{2}$,

$\mathcal{L}_{m}:=\begin{cases}\text{Sym
}H_{m-1}\big{(}\text{conf}(m,w);\mathbb{Q}\big{)}\cong\text{Sym
}H_{m-1}\big{(}F_{m}(\mathbb{R}^{2});\mathbb{Q}\big{)}&\mbox{for $m$ odd}\\\
\bigwedge H_{m-1}\big{(}\text{conf}(m,w);\mathbb{Q}\big{)}\cong\bigwedge
H_{m-1}\big{(}F_{m}(\mathbb{R}^{2});\mathbb{Q}\big{)}&\mbox{for $m$
even}\end{cases}$

is a twisted (skew-)commutative algebras over $\mathbb{Q}$. If $m=1$, this tca
is isomorphic to $\text{Sym}\big{(}\text{Sym}^{1}(\mathbb{Q})\big{)}$, and if
$m=2$, it is isomorphic to
$\bigwedge\big{(}\text{Sym}^{2}(\mathbb{Q})\big{)}$. See [SS12] and [NSS19]
for more information on these twisted commutative algebras.

######  Definition 4.8.

A _module over a tca $A$_ is a graded $A$-module (in the usual sense) equipped
with an action of $S_{n}$ on $M_{n}$ for all $n$ for which the multiplication
map

$A_{n}\otimes M_{m}\to M_{n+m}$

is $S_{n}\times S_{m}$ equivariant.

######  Example 4.9.

Being a module over the tca $R$ is equivalent to being an FB-module over $R$,
where FB is the category of finite sets and bijections. In an abuse of
notation, we write FB-mod for the category of modules over $R$ when $R$ is
viewed as tca. For more on FB, see, for example, [CEF15].

######  Example 4.10.

If $X$ is a connected noncompact finite type manifold of dimension $d\geq 2$,
then for all $i\geq 0$, $H_{i}\big{(}F_{\bullet}(X)\big{)}$ is a module over
$\mathcal{L}_{1}$. Here multiplication of an element $Y\in
H_{i}\big{(}F_{m}(X)\big{)}$ by the fundamental class $Z$ of
$H_{i}\big{(}F_{1}(\mathbb{R}^{d})\big{)}$ is $Z\otimes Y\in
H_{i}\big{(}F_{m+1}(X)\big{)}$. See [MW19, Section 1.1].

######  Definition 4.11.

Let $M$ be a module over a tca $A$. Let _$H^{A}_{0}(M)_{S}$_ be the quotient

$H^{A}_{0}(M)_{S}:=\text{coker}\Big{(}\bigoplus_{S=P\sqcup
Q,P\neq\emptyset}A_{P}\otimes M_{Q}\to M_{S}\Big{)}.$

######  Definition 4.12.

We say that an $A$-module $M$ is _finitely generated_ if
$\bigoplus^{\infty}_{k=0}H^{A}_{0}(M)_{k}$ is finitely generated as an
$R$-module.

######  Definition 4.13.

Let $M$ be an module over the tca $A$. We say that $\deg M\leq d$ if $M_{k}=0$
for all $k>d$. We say that $M$ is _generated in degree $\leq d$_ if $\deg
H^{A}_{0}(M)\leq d$.

We will see that the twisted algebra
$H_{*}\big{(}\text{conf}(\bullet,w);\mathbb{Q}\big{)}$ is a finitely generated
module over a number of twisted commutative algebras.

######  Definition 4.14.

Let $A$ be a tca. We define

$M^{A}:\text{FB-mod}\to A\text{-mod}$

via the formula

$M^{A}(U):=A\otimes_{\text{FB}}U,$

where $U$ is an FB-mod. The $A$-module structure on $M^{A}(U)$ is induced by
the $A\otimes A\to A$. An $A$-module of the form $M^{A}(U)$ is said to be
_free_.

If $M$ is a finitely generated module over the tca $A$, we say that it is
representation stable.

######  Example 4.15.

Church–Ellenberg–Farb and Miller–Wilson proved that if $X$ is a connected
noncompact finite type manifold of dimension at least $2$, then
$H_{k}\big{(}F_{\bullet}(X);R\big{)}$ is a finitely generated free
$\text{Sym}\Big{(}H_{0}\big{(}F_{1}(\mathbb{R}^{d})\big{)}\Big{)}$-module
generated in degree $\leq 2k$ for all $k$ [CEF15, MW19, Theorem 6.4.3, Theorem
1.1].

We will need slightly larger twisted commutative algebras than
$\mathcal{L}_{m}$ to prove first (and higher) order representation stability
results for the ordered configuration space of open unit-diameter disks in the
infinite strip of width $w$.

######  Definition 4.16.

For $m\leq\frac{w}{2}$, let

$\mathcal{L}^{w}_{m}(b):=\begin{cases}\text{Sym
}\bigg{(}\Big{(}H_{m-1}\big{(}\text{conf}(m,w);\mathbb{Q}\big{)}\Big{)}^{\oplus
b}\bigg{)}&\mbox{for $m$ odd}\\\
\bigwedge\bigg{(}\Big{(}H_{m-1}\big{(}\text{conf}(m,w);\mathbb{Q}\big{)}\Big{)}^{\oplus
b}\bigg{)}&\mbox{for $m$ even}\end{cases}.$

When $b=1$, these $\mathcal{L}^{w}_{m}$ is isomorphic to $\mathcal{L}_{m}$,
the twisted commutative algebras that Miller–Wilson suggested could be used to
formalize higher order representation stability for the homology of ordered
configuration space of points in a connected noncompact manifold of dimension
$d\geq 2$, as $F_{m}(\mathbb{R}^{2})\simeq\text{conf}(m,w)$ and wheels on $m$
disks (anti)-commute. See [MW19, Definition 3.29].

Finally, we state a proposition that we will need later when proving our
representation stability results.

######  Proposition 4.17.

Let $M$ an $\mathcal{L}^{w}_{n}(b)$-module. Then $M$ is also an
$\mathcal{L}^{w}_{n}(b+1)$-module.

###### Proof.

Identify multiplication by an element in the $(b+1)^{\text{th}}$ copy of
$H_{n-1}\big{(}\text{conf}(n,w);\mathbb{Q}\big{)}$ by multiplication by an
isomorphic element in the $b^{\text{th}}$-copy of
$H_{n-1}\big{(}\text{conf}(n,w);\mathbb{Q}\big{)}$, i.e., if $X_{b+1}$ is in
the $(b+1)^{\text{th}}$ copy of
$H_{n-1}\big{(}\text{conf}(n,w);\mathbb{Q}\big{)}$ and $X_{b}\cong X_{b+1}$ is
in the $b^{\text{th}}$-copy of
$H_{n-1}\big{(}\text{conf}(n,w);\mathbb{Q}\big{)}$, then for all $m\in M$ set
$X_{b+1}\otimes m=X_{b}\otimes m$. ∎

For more on twisted commutative algebras and their connection to
representation stability see [SS12, SS17, NSS19].

###  4.2 First-Order Representation Stability for the Homology of
$\text{conf}(\bullet,w)$

We prove that for all $k\geq 0$ and all $w\geq 2$, the homology groups
$H_{k}\big{(}\text{conf}(\bullet,w);\mathbb{Q}\big{)}$ have the structure of a
finitely generated $\mathcal{L}^{w}_{1}(b+1)$-module, where $b$ depends only
on $k$ and $w$, i.e., that the ordered configuration space of open unit-
diameter disks on the infinite strip of width $w$ is first-order
representation stable. First, we recall a similar result of Alpert for
$H_{k}\big{(}\text{conf}(\bullet,2);\mathbb{Z}\big{)}$ and a proposition of
Alpert–Manin that suggests Alpert’s technique does not generalize to larger
$w$.

######  Definition 4.18.

Let $Z\in H_{k}\big{(}\text{conf}(n,w);\mathbb{Z}\big{)}$ be a concatenation
product of wheels and filters, then an _integral barrier_ in $Z$ is any wheel
on $w$ disks or any filter.

When $w=2$ every element in $H_{k}\big{(}\text{conf}(n,w);\mathbb{Z}\big{)}$
has exactly $k$ integral barriers. Alpert proved that for all $k\geq 0$, the
homology groups $H_{k}\big{(}\text{conf}(\bullet,2);\mathbb{Z}\big{)}$ have
the structure of a finitely generated $\mathcal{L}^{w}_{1}(k+1)$-module over
$\mathbb{Z}$ generated in degree $\leq 3k$ [Alp20, Theorem 6.1].
Multiplication of a class $Z\in
H_{k}\big{(}\text{conf}(n,2);\mathbb{Z}\big{)}$ by the generator for the
$i^{\text{th}}$ copy of $H_{0}\big{(}\text{conf}(1,2);\mathbb{Z}\big{)}$ in
$\mathcal{L}^{w}_{1}(k+1)$, corresponds to inserting a wheel on $1$ disk
immediately to the left of the $i^{\text{th}}$ barrier of $Z$ if $i\leq k$,
and to inserting a wheel on $1$ disk at the far right of $Z$ if $i=k+1$. See
Figure 23.

Figure 23: A class in the basis for
$H_{2}\big{(}\text{conf}(5,2);\mathbb{Z}\big{)}$ from Theorem 3.18 consisting
of a concatenation product of a proper wheel and a proper filter. The
$\mathcal{L}^{2}_{1}(3)$-module structure of
$H_{2}\big{(}\text{conf}(\bullet,2);\mathbb{Z}\big{)}$ arises from the fact
that there are 3 regions in which one could insert a wheel on $1$ disk, namely
to the left of wheel, in-between the wheel and the filter, and to the right of
the filter.

Alternatively, one could restate Alpert’s theorem by saying that
$H_{k}\big{(}\text{conf}(\bullet,2);\mathbb{Z}\big{)}$ is a finitely generated
FIk+1-module over $\mathbb{Z}$ generated in degree $\leq 3k$. For more on FId-
modules and their role in representation stability, see [Ram17].

Unfortunately, Alpert’s theorem does not generalize to
$H_{k}\big{(}\text{conf}(\bullet,w);\mathbb{Z}\big{)}$ for $w\geq 3$ as the
operation of counting barriers is not well defined for $w\geq 3$, as evinced
by the following proposition of Alpert–Manin.

######  Proposition 4.19.

(Alpert–Manin [AM21, Proposition 8.6]) There is a nontrivial element $Z\in
H_{4}(\text{conf}(8,3);\mathbb{Z})$ that can be written as a sum of products
of 4 proper wheels each of which consists of 2 disks, or as a sum of products
of 2 proper wheels each consisting of 2 disks and a proper filter consisting
of 4 disks, or as a sum of products of 2 proper filters each consisting of 4
disks.

In general, for all $w\geq 3$ and $n\gg 0$, there are relations between
classes with different numbers of integral barriers as in Proposition 4.19. It
follows that the number of integral barriers in a homology class is not well-
defined, and as such one can not extend Alpert’s proof of first-order
representation stability to $\text{conf}(\bullet,w)$ for $w\geq 3$. We will
show that there is a better notion of barriers when we consider rational
homology and use the generating set for homology given in the proof of Theorem
4.4.

######  Definition 4.20.

Let $Z\in H_{k}\big{(}\text{conf}(n,w);\mathbb{Z}\big{)}$ be a concatenation
product of wheels and nontrivial averaged-filters, then for $1\leq
d\leq\big{\lfloor}\frac{w}{2}\big{\rfloor}$, a _rational $d$-barrier_ in $Z$
is any wheel on $w+1-d$ disks or any averaged-filter.

We write $B(n,k,w,d,m)$ for the subspace of elements in
$H_{k}\big{(}\text{conf}(n,w);\mathbb{Q}\big{)}$ that can be written as a
linear combination of elements with $m$ rational $d$-barriers. We will show
that the $B(n,k,w,d,m)$ are well-defined, and if $m\neq m^{\prime}$, then then
subspaces $B(n,k,w,d,m)$ and $B(n,k,w,d,m^{\prime})$ have trivial
intersection.

######  Proposition 4.21.

For all $w\geq 2$, $0\leq d\leq\lfloor\frac{w}{2}\rfloor$, and $k,n\geq 0$

$H_{k}\big{(}\text{conf}(n,w);\mathbb{Q}\big{)}=\bigoplus_{m}B(n,k,w,d,m).$

Moreover, $B(n,k,w,d,m)$ is $S_{n}$-invariant.

###### Proof.

Let $Z\in H_{k}\big{(}\text{conf}(n,w);\mathbb{Q}\big{)}$ be a product of
generators from the presentation for
$H_{*}\big{(}\text{conf}(\bullet,w);\mathbb{Q}\big{)}$ given in Theorem 4.4,
i.e., $Z=Z_{1}|\cdots|Z_{r}$, where $Z_{i}$ is either a proper wheel such that
the largest label comes first, or a proper averaged-filter on at least $3$
wheels such that the wheels are in order of increasing rank and in each wheel
the largest label comes first. By definition, the number of rational
$d$-barriers in $Z$ is the number of $Z_{i}$ that are either proper wheels on
at least $w+1-d$ disks or proper averaged-filters. The $S_{n}$-action
preserves the number of proper wheels and proper averaged-filters and their
sizes, as the symmetric group sends a proper wheel on $l$ disks to a proper
wheel on $l$ disks and a proper averaged-filter on $p$ wheels to a proper
averaged-filter on $p$ wheels, each of the same size. The $5$ types of
relations of the presentation for
$H_{*}\big{(}\text{conf}(\bullet,w);\mathbb{Q}\big{)}$ given in the proof of
Theorem 4.4 preserve the number of proper wheels on at least $\frac{w}{2}+1$
disks not in an averaged-filter as well as the number of proper averaged-
filters. Since $d\leq\frac{w}{2}$, it follows that $n+1-d\geq\frac{w}{2}+1$.
Therefore, the number of rational $d$-barriers in an element of
$H_{k}\big{(}\text{conf}(n,w);\mathbb{Q}\big{)}$ is preserved by the action of
$S_{n}$. Moreover, this proves that the intersection between $B(n,k,w,d,m)$
and $B(n,k,w,d,m^{\prime})$ is trivial if $m\neq m^{\prime}$. It follows that
$B(n,k,w,d,m)$ is $S_{n}$-invariant and

$H_{k}\big{(}\text{conf}(n,w);\mathbb{Q}\big{)}=\bigoplus_{m}B(n,k,w,d,m).$

∎

It follows from Proposition 4.21 that counting the number of rational
$d$-barriers is a well-defined operation. Additionally, this proposition
proves that we can take the basis elements for
$H_{k}\big{(}\text{conf}(n,w);\mathbb{Q}\big{)}$ from Theorem 3.34 with $m$
rational $d$-barriers to be a basis for $B(n,k,w,d,m)$, as each basis element
has a fixed number of barriers. We use this to prove that the ordered
configuration space of unit diameter disks in the infinite strip satisfies a
notion of first-order representation stability.

######  Theorem 4.22.

For all $k\geq 0$ and $w\geq 2$, the homology groups
$H_{k}\big{(}\text{conf}(\bullet,w);\mathbb{Q}\big{)}$ have the structure of
an $\mathcal{L}^{w}_{1}(b+1)$-module, where
$b=\big{\lfloor}\frac{k}{w-1}\big{\rfloor}$, finitely generated in degree
$\leq 2k$ for $w\geq 3$, and degree $\leq 3k$ for $w=2$.

###### Proof.

We begin by proving that
$H_{k}\big{(}\text{conf}(\bullet,w);\mathbb{Q}\big{)}$ is an
$\mathcal{L}^{w}_{1}(b+1)$-module, where
$b=\big{\lfloor}\frac{k}{w-1}\big{\rfloor}$. Then we show that this
$\mathcal{L}^{w}_{1}(b+1)$-module is finitely generated through a quick
counting argument.

Let $X_{i}\in\mathcal{L}^{w}_{1}(b+1)$ denote the fundamental class of the
$i^{\text{th}}$ copy of $H_{0}\big{(}\text{conf}(1,w);\mathbb{Q}\big{)}$ in
$\mathcal{L}^{w}_{1}(b+1)$, and let $Z$ be an element in the basis for
$H_{k}\big{(}\text{conf}(n,w);\mathbb{Q}\big{)}$ from Theorem 3.34 such that
$Z$ has $m$ rational $1$-barriers. For $i\leq m$, set $X_{i}\otimes Z$ to be
element of $H_{k}\big{(}\text{conf}(n+1,w);\mathbb{Q}\big{)}$ arising from
inserting a wheel on $1$ disk labeled $n+1$ immediately to the left of the
$i^{\text{th}}$ rational $1$-barrier of $Z$, and for $i>m$, set $X_{i}\otimes
Z$ to be the element of $H_{k}\big{(}\text{conf}(n+1,w);\mathbb{Q}\big{)}$
arising from $Z$ by placing a wheel on $1$ disk labeled $n+1$ to the right of
$Z$. By Proposition 4.21 this is well defined.

Since $w\geq 2$, Proposition 3.21 proves that two wheels each consisting of
$1$ disk commute. Therefore, the multiplication map

$\big{(}\mathcal{L}^{w}_{1}(b+1)\big{)}_{m}\otimes
H_{k}(\text{conf}(n,w);\mathbb{Q})\to H_{k}(\text{conf}(n+m,w);\mathbb{Q})$

is $S_{m}\times S_{n}$ invariant, proving that
$H_{k}\big{(}\text{conf}(\bullet,w);\mathbb{Q}\big{)}$ has the structure of an
$\mathcal{L}^{w}_{1}(b+1)$-module.

To see that as an $\mathcal{L}^{w}_{1}(b+1)$-module
$H_{k}\big{(}\text{conf}(\bullet,w);\mathbb{Q}\big{)}$ is finitely generated
in degree $\leq 2k$ for $w\geq 3$ and degree $\leq 3k$ for $w=2$, note that
there can be at most $b=\big{\lfloor}\frac{k}{w-1}\big{\rfloor}$ rational
$1$-barriers in a basis element of $H_{k}(\text{conf}(\bullet,w);\mathbb{Q})$.
This follows from the fact that the $1$-barriers using the fewest disks are
the wheels consisting of $w$ disks, and these wheel have homological degree
$w-1$. It follows that an element of
$H_{k}\big{(}\text{conf}(\bullet,w);\mathbb{Q}\big{)}$ can have at most
$\lfloor\frac{k}{w-1}\rfloor$ such $1$-barriers.

In every element of the basis of Theorem 3.34, the only place wheels on $1$
disk not in an averaged-filter are allowed is immediately to the left of a
$1$-barrier or at the far right. Therefore, an element $Z$ in
$H_{k}\big{(}\text{conf}(n,w);\mathbb{Q}\big{)}$ that can be written as
product of proper wheels and proper averaged-filters on at least $3$ wheels,
such that one of the wheels consists of a single disk arises from a nontrivial
multiplication of $\big{(}\mathcal{L}^{w}_{1}(b+1)\big{)}_{1}$ on an element
of $H_{k}\big{(}\text{conf}(n-1,w);\mathbb{Q}\big{)}$.

The generators of $H_{k}\big{(}\text{conf}(\bullet,w);\mathbb{Q}\big{)}$ that
have the worst ratio of number of disks to homological degree are the wheels
on $2$ disks for $w\geq 3$, and averaged-filters on $3$ wheels, each
consisting of $1$ disk for $w=2$. Each of these generators have homological
dimension $1$. It follows that if $w\geq 3$ and $n>2k$, then every element of
$H_{k}\big{(}\text{conf}(n,w);\mathbb{Q}\big{)}$ can be written as a sum of
elements each of which has at least $1$ wheel on $1$ disk outside of an
averaged-filter. The same holds for $w=2$ and $n>3k$. Therefore,

$H^{\mathcal{L}_{1}(b+1)}_{0}\Big{(}H_{k}\big{(}\text{conf}(\bullet,w);\mathbb{Q}\big{)}\Big{)}_{n}=0$

for all $n>2k$ when $w\geq 3$ and $n>3k$ when $w=2$, bounding the generation
degree of $H_{k}\big{(}\text{conf}(\bullet,w);\mathbb{Q}\big{)}$ as an
$\mathcal{L}^{w}_{1}(b+1)$-module ∎

It follows that Theorem 4.22 proves that the ordered configuration space of
unit-diameter disks in the infinite strip of width $w$ is first-order
representation stable.

Theorem 4.22 is equivalent to the fact that for $k\geq 0$ and $w\geq 2$, the
homology groups $H_{k}\big{(}\text{conf}(\bullet,w);\mathbb{Q}\big{)}$ have
the structure of an FIb+1-module, where
$b=\big{\lfloor}\frac{k}{w-1}\big{\rfloor}$, finitely generated in degree $2k$
for $w\geq 3$, and degree $3k$ for $w=2$, i.e., Theorem 1.2. Thus, we have
generalized Theorem 6.1 of Alpert in the case of rational homology and all
widths $w\geq 2$ [AM21, Theorem 6.1]. By Ramos’s results on FId-modules, it
follows that one can use Theorem 4.22 to get upper bounds for Betti numbers of
$H_{k}\big{(}\text{conf}(n,w);\mathbb{Q}\big{)}$ for $n>2k$ if $w\geq 3$ and
$n>3k$ if $w=2$ if one knows the Betti numbers for
$H_{k}\big{(}\text{conf}(m,w);\mathbb{Q}\big{)}$ for all $m\leq 2k$ if $w\geq
3$ and $m\leq 3k$ if $w=2$. Additionally, one gets upper bounds for the
multiplicities of irreducible $S_{n}$-representations in a decomposition of
$H_{k}\big{(}\text{conf}(n,w);\mathbb{Q}\big{)}$ for $n>2k$ if $w\geq 3$ and
$n>3k$ if $w=2$ if one how to decompose
$H_{k}\big{(}\text{conf}(m,w);\mathbb{Q}\big{)}$ into irreducible
$S_{m}$-representations for all $m\leq 2k$ if $w\geq 3$ and $m\leq 3k$ if
$w=2$.

In the next section, we will show that the ordered configuration space of
unit-diameter disks in the infinite strip of width $w$ exhibits notions of
higher order representation stability.

###  4.3 Higher Order Stability

In this section, we prove that the ordered configuration space of open unit-
diameter disks in the infinite strip of width $w$ exhibits reasonable notions
of higher order representation stability. In the previous section, we proved
that $H_{k}\big{(}\text{conf}(\bullet,w);\mathbb{Q}\big{)}$ is a finitely
generated module over the twisted commutative algebra
$\mathcal{L}^{w}_{1}(b+1)$ where $b=\big{\lfloor}\frac{k}{w-1}\big{\rfloor}$.
Note as $k$ varies, $\mathcal{L}^{w}_{1}(b+1)$ varies. This is different from
the case of first-order representation stability for configuration spaces of
points in connected noncompact manifolds, where the twisted algebra of first-
order stability is always isomorphic to $\mathcal{L}_{1}$, regardless of the
value of $k$. In their statement of second-order representation stability for
the ordered configuration space of points in a connected noncompact manifold,
Miller–Wilson used the zeroth $\mathcal{L}_{1}$-homology of the homology of
configuration space. Unfortunately, we cannot do the same for finite $b$.
However, by taking a limit we are able to remedy this without losing any
information.

Recall that Proposition 4.17 proves that any module $M$ over
$\mathcal{L}^{w}_{1}(b)$ can be made a module over $\mathcal{L}^{w}_{1}(b+1)$
by setting $X_{b+1}\otimes m=X_{b}\otimes m$ for all $m\in M$, where $X_{j}$
is the fundamental class of the $j^{\text{th}}$ copy of
$H_{0}\big{(}\text{conf}(1,w);\mathbb{Q}\big{)}$ in $\mathcal{L}^{w}_{1}(b)$.
By setting $X_{i}\otimes m=X_{b+1}\otimes m$ for all $i>b+1$ and all $m\in M$
and where $b=\big{\lfloor}\frac{k}{w-1}\big{\rfloor}$, it follows that for all
$k$ and $w$ we have that
$H_{k}\big{(}\text{conf}(\bullet,w);\mathbb{Q}\big{)}$ is an
$\mathcal{L}^{w}_{1}(\infty)$-module, and

$H^{\mathcal{L}^{w}_{1}(\infty)}_{0}\Big{(}H_{k}\big{(}\text{conf}(\bullet,w);\mathbb{Q}\big{)}\Big{)}=H^{\mathcal{L}^{w}_{1}(b+1)}_{0}\Big{(}H_{k}\big{(}\text{conf}(\bullet,w);\mathbb{Q}\big{)}\Big{)}.$

As a corollary to Theorem 4.22, for all $k\geq 0$ and $w\geq 2$, the homology
groups $H_{k}\big{(}\text{conf}(\bullet,w);\mathbb{Q}\big{)}$ have the
structure of an $\mathcal{L}^{w}_{1}(\infty)$-module, finitely generated in
degree $2k$ for $w\geq 3$ and degree $3k$ for $w=2$.

It follows that the twisted algebra
$H_{*}\big{(}\text{conf}(\bullet,w);\mathbb{Q}\big{)}$ is an
$\mathcal{L}^{w}_{1}(\infty)$-module, though it is not finitely generated as
one as we need generators in each degree $k$. Still, we can make sense of the
the zeroth $\mathcal{L}^{w}_{1}(\infty)$-homology of
$H_{*}(\text{conf}(\bullet,w);\mathbb{Q})$.

######  Proposition 4.23.

For all $w\geq 2$,

$H^{\mathcal{L}^{w}_{1}(\infty)}_{0}\Big{(}H_{*}\big{(}\text{conf}(\bullet,w);\mathbb{Q}\big{)}\Big{)}$

is a finitely presented twisted algebra. Moreover, there is a presentation for
$H^{\mathcal{L}^{w}_{1}(\infty)}_{0}\Big{(}H_{*}\big{(}\text{conf}(\bullet,w);\mathbb{Q}\big{)}\Big{)}$
with the same generators and relations as
$H_{*}\big{(}\text{conf}(\bullet,w);\mathbb{Q}\big{)}$ has in the proof of
Theorem 4.4, along the extra relation that wheels on $1$ disk not in an
averaged-filter are trivial.

###### Proof.

First, note that
$H^{\mathcal{L}^{w}_{1}(\infty)}_{0}\Big{(}H_{*}\big{(}\text{conf}(\bullet,w);\mathbb{Q}\big{)}\Big{)}$
is a twisted algebra as it is an associative graded unital
$\mathbb{Q}$-algebra with an $S_{n}$-action on
$H^{\mathcal{L}^{w}_{1}(\infty)}_{0}\Big{(}H_{*}\big{(}\text{conf}(n,w);\mathbb{Q}\big{)}\Big{)}$
for all $n$, and multiplication of an element in
$H^{\mathcal{L}^{w}_{1}(\infty)}_{0}\Big{(}H_{k}\big{(}\text{conf}(n,w);\mathbb{Q}\big{)}\Big{)}$
by an element in
$H^{\mathcal{L}^{w}_{1}(\infty)}_{0}\Big{(}H_{j}\big{(}\text{conf}(m,w);\mathbb{Q}\big{)}\Big{)}$
is $S_{n}\times S_{m}$ invariant.

Let $X_{i}\in\big{(}\mathcal{L}^{w}_{1}(\infty)\big{)}_{1}$ be the fundamental
class of the $i^{\text{th}}$ copy of
$H_{0}\big{(}\text{conf}(1,w);\mathbb{Q}\big{)}$ in
$\mathcal{L}^{w}_{1}(\infty)$. Multiplication by $X_{i}$ corresponds inserting
a wheel on $1$ disk immediately to the left of the $i^{\text{th}}$ $1$-barrier
for small $i$ and at the far right for large $i$. By applications of the
(anti)-commutation relations of Proposition 3.21 it follows that
multiplication by a wheel on $1$ disk in
$H_{*}\big{(}\text{conf}(\bullet,w);\mathbb{Q}\big{)}$ corresponds to
multiplication by $X_{i}$ for some $i$. Therefore,
$H^{\mathcal{L}^{w}_{1}(\infty)}_{0}\Big{(}H_{*}\big{(}\text{conf}(\bullet,w);\mathbb{Q}\big{)}\Big{)}$
is a twisted algebra with the same generators and relations as
$H_{*}\big{(}\text{conf}(\bullet,w);\mathbb{Q}\big{)}$ has in Theorem 4.4
along with the extra relation that wheels on $1$ disk are trivial. ∎

We prove two results that will help us prove that the ordered configuration
space of unit diameter disks in the infinite strip of width $w$ exhibits
notions of $2^{\text{nd}}-$ through
$\Big{(}\big{\lfloor}\frac{w}{2}\big{\rfloor}\Big{)}^{\text{th}}$-order
representation stability.

######  Proposition 4.24.

For $0\leq d\leq\big{\lfloor}\frac{w}{2}\big{\rfloor}$, the homology groups
$H_{*}\big{(}\text{conf}(\bullet,w);\mathbb{Q}\big{)}$ have the structure of
an $\mathcal{L}^{w}_{d}(\infty)$-module.

###### Proof.

For fixed $d$, we can used Proposition 4.21 to decompose
$H_{k}\big{(}\text{conf}(n,w);\mathbb{Q}\big{)}$:

$H_{k}\big{(}\text{conf}(n,w);\mathbb{Q}\big{)}=\bigoplus_{m}B(n,k,w,d,m).$

By Proposition 3.19 the $S_{d}$-span of the proper wheel $W(d,d-1,\dots,1)$
generates $H_{d-1}\big{(}\text{conf}(d,w);\mathbb{Q}\big{)}$. Let
$X_{i}\in\mathcal{L}^{w}_{d}(\infty)$ correspond to the wheel
$W(d,d-1,\dots,1)$ in the $i^{\text{th}}$-copy of
$H_{d-1}\big{(}\text{conf}(d,w);\mathbb{Q}\big{)}$ in
$\mathcal{L}^{w}_{d}(\infty)$, and let multiplication of $Z\in
B(n,k,w,d,m)\subseteq H_{k}\big{(}\text{conf}(n,w);\mathbb{Q}\big{)}$ by
$X_{i}$ correspond to placing a wheel labeled $W(n+d,n+d-1,\dots,n+1)$ to the
immediate left of the $i^{\text{th}}$ rational $d$-barrier of $z$ if $i\leq
m$, and to the far right if $i>m$. This is an $S_{d}\times S_{n}$-equivariant
map

$\mathcal{L}^{w}_{d}(\infty)_{d}\otimes
H_{k}\big{(}\text{conf}(n,w);\mathbb{Q}\big{)}\to
H_{k+d-1}\big{(}\text{conf}(n+d,w);\mathbb{Q}\big{)},$

and this hold for all $n,k\geq 0$. Since the $X_{i}$ generate
$\mathcal{L}^{w}_{d}(\infty)$ it follows that this defines an
$\mathcal{L}^{w}_{d}(\infty)$-module structure on
$H_{*}\big{(}\text{conf}(\bullet,w);\mathbb{Q}\big{)}$ ∎

Next, we shows that this $\mathcal{L}^{w}_{d}(\infty)$-module structure
extends to a series of quotient of
$H_{*}\big{(}\text{conf}(\bullet,w);\mathbb{Q}\big{)}$.

######  Lemma 4.25.

For $0\leq d\leq\lfloor\frac{w}{2}\rfloor$, the quotient

$H^{\mathcal{L}^{w}_{d}(\infty)}_{0}\Bigg{(}\cdots\bigg{(}H^{\mathcal{L}^{w}_{1}(\infty)}_{0}\Big{(}H_{*}\big{(}\text{conf}(\bullet,w);\mathbb{Q}\big{)}\Big{)}\bigg{)}\cdots\Bigg{)}$

is a finitely presented twisted algebra. Moreover, it has a finite
presentation consisting of the generators and the relations given for

$H_{*}\big{(}\text{conf}(\bullet,w);\mathbb{Q}\big{)}$

in the proof of Theorem 4.4 along with the relations that any proper wheel on
no more than $d$ disks that is not in an averaged-filter is trivial.

###### Proof.

The case $d=0$ is Theorem 4.4, and the case $d=1$ is Proposition 4.23. Assume
that we have proved the lemma through the case for $d\leq j$, for some
$j\leq\frac{w}{2}-1$. We will show that the Lemma holds in the case $d=j+1$.

Note that

$H^{\mathcal{L}^{w}_{j+1}(\infty)}_{0}\Bigg{(}\cdots\bigg{(}H^{\mathcal{L}^{w}_{1}(\infty)}_{0}\Big{(}H_{*}\big{(}\text{conf}(\bullet,w);\mathbb{Q}\big{)}\Big{)}\bigg{)}\cdots\Bigg{)}$

satisfies the definition of a twisted algebra for the same reasons

$H^{\mathcal{L}^{w}_{1}(\infty)}_{0}\Big{(}H_{*}\big{(}\text{conf}(\bullet,w);\mathbb{Q}\big{)}\Big{)}$

does.

Since we have assumed that the Lemma holds in the case $d=j$, we have that

$H^{\mathcal{L}^{w}_{j}(\infty)}_{0}\Bigg{(}\cdots\bigg{(}H^{\mathcal{L}^{w}_{1}(\infty)}_{0}\Big{(}H_{*}\big{(}\text{conf}(\bullet,w);\mathbb{Q}\big{)}\Big{)}\bigg{)}\cdots\Bigg{)}$

is a finitely presented twisted algebra, and it has a presentation consisting
of the same generators and relations as the presentation for
$H_{*}\big{(}\text{conf}(\bullet,w);\mathbb{Q}\big{)}$ found in the proof of
Theorem 4.4 along with the relations that all proper wheels on at most $j$
disks not inside an averaged-filter are trivial. Note, these extra relations
preserve the number of rational $j+1$-barriers as only wheels on more than
$n-j>\frac{w}{2}$ disks can act as rational $j+1$-barriers. Thus, the image of
$B(n,k,w,j+1,m)\subset H_{k}\big{(}\text{conf}(n,w);\mathbb{Q}\big{)}$ under
the quotient is well-defined for all $n$, $k$, and $m$, and multiplication by
$X_{i}\in\mathcal{L}^{w}_{j+1}(\infty)$, where $x$ is a proper wheel on $j+1$
disks in the $i^{\text{th}}$-copy of
$H_{j}\big{(}\text{conf}(j+1,w);\mathbb{Q}\big{)}$ is well-defined.

Every element of
$H^{\mathcal{L}^{w}_{j}(\infty)}_{0}\Bigg{(}\cdots\bigg{(}H^{\mathcal{L}^{w}_{1}(\infty)}_{0}\Big{(}H_{*}\big{(}\text{conf}(\bullet,w);\mathbb{Q}\big{)}\Big{)}\bigg{)}\cdots\Bigg{)}$
can be written as sum of products of generators such that any wheel on $j+1$
disks is part of a sequence of consecutive wheels on $j+1$ disks immediately
to the right of a rational $j+1$-barrier or at the far right of the product.
This follows since the relations given in the presentation in the proof of
Theorem 4.4 allow us to (anti)-commute all smaller wheels as
$j+1\leq\frac{w}{2}$ and all smaller wheels are trivial by the induction
hypothesis. Therefore, multiplication by
$X_{i}\in\mathcal{L}^{w}_{j+1}(\infty)$ corresponds to multiplication in
$H^{\mathcal{L}^{w}_{j}(\infty)}_{0}\Bigg{(}\cdots\bigg{(}H^{\mathcal{L}^{w}_{1}(\infty)}_{0}\Big{(}H_{*}\big{(}\text{conf}(\bullet,w);\mathbb{Q}\big{)}\Big{)}\bigg{)}\cdots\Bigg{)}$
by a wheel on $j+1$ disks, and the converse holds as well.

Thus, the quotient

$H^{\mathcal{L}^{w}_{j+1}(\infty)}_{0}\Bigg{(}\cdots\bigg{(}H^{\mathcal{L}^{w}_{1}(\infty)}_{0}\Big{(}H_{*}\big{(}\text{conf}(\bullet,w);\mathbb{Q}\big{)}\Big{)}\bigg{)}\cdots\Bigg{)}$

has a finite presentation consisting of the same generators and relations we
gave for $H_{*}\big{(}\text{conf}(\bullet,w);\mathbb{Q}\big{)}$ in the proof
of Theorem 4.4, along with the extra relations that all wheels on no more than
$j+1$ disks are trivial. ∎

We use Lemma 4.25 to prove that the ordered configuration space of unit
diameter disks in the infinite strip of width $w$ exhibits notions of
$2^{\text{nd}}$\- through
$\big{\lfloor}\frac{w}{2}\big{\rfloor}^{\text{th}}$-order representation
stability. First, we prove an intermediate proposition.

######  Proposition 4.26.

Let

$\mathcal{W}^{d}_{i}(\bullet):=H^{\mathcal{L}^{w}_{d-1}(\infty)}_{0}\Bigg{(}\cdots\bigg{(}H^{\mathcal{L}^{w}_{1}(\infty)}_{0}\Big{(}H_{\frac{(d-1)|\bullet|+i}{d}}\big{(}\text{conf}(\bullet,w);\mathbb{Q}\big{)}\Big{)}\bigg{)}\cdots\Bigg{)}.$

For $1\leq d\leq\big{\lfloor}\frac{w}{2}\big{\rfloor}$, the sequence
$\mathcal{W}^{d}_{i}(\bullet)$ has the structure of a finitely generated
$\mathcal{L}^{w}_{d}(\infty)$-module generated in degree $(d+1)i$ for
$2d+1\leq w$, and degree $(d+1)i+d$ for $2d=w$.

###### Proof.

By Lemma 4.25,

$H^{\mathcal{L}^{w}_{d-1}(\infty)}_{0}\Bigg{(}\cdots\bigg{(}H^{\mathcal{L}^{w}_{1}(\infty)}_{0}\Big{(}H_{*}\big{(}\text{conf}(\bullet,w);\mathbb{Q}\big{)}\Big{)}\bigg{)}\cdots\Bigg{)}$

is an $\mathcal{L}^{w}_{d}(\infty)$-module. Since multiplication by a
generator of $\mathcal{L}^{w}_{d}(\infty)$ increases homological degree by
$d-1$ and the number of disks by $d$, it follows that

$H^{\mathcal{L}^{w}_{d-1}(\infty)}_{0}\Bigg{(}\cdots\bigg{(}H^{\mathcal{L}^{w}_{1}(\infty)}_{0}\Big{(}H_{*}\big{(}\text{conf}(\bullet,w);\mathbb{Q}\big{)}\Big{)}\bigg{)}\cdots\Bigg{)}$

decomposes as a direct sum of $\mathcal{L}^{w}_{d}(\infty)$-modules of the
form $\mathcal{W}^{d}_{i}(\bullet)$, i.e.,

$H^{\mathcal{L}^{w}_{d-1}(\infty)}_{0}\Bigg{(}\cdots\bigg{(}H^{\mathcal{L}^{w}_{1}(\infty)}_{0}\Big{(}H_{*}\big{(}\text{conf}(\bullet,w);\mathbb{Q}\big{)}\Big{)}\bigg{)}\cdots\Bigg{)}\\\
=\bigoplus_{i=0}^{\infty}H^{\mathcal{L}^{w}_{d-1}(\infty)}_{0}\Bigg{(}\cdots\bigg{(}H^{\mathcal{L}^{w}_{1}(\infty)}_{0}\Big{(}H_{\frac{(d-1)|\bullet|+i}{d}}\big{(}\text{conf}(\bullet,w);\mathbb{Q}\big{)}\Big{)}\bigg{)}\cdots\Bigg{)}.$

We count the maximal number of rational $d$-barriers in
$\mathcal{W}^{d}_{i}(\bullet)$ as this will give us a bound on the generation
degree. The maximal number of rational $d$-barriers in
$\mathcal{W}^{d}_{i}(\bullet)$ is equal to the largest nontrivial term of the
sequence
$H^{\mathcal{L}^{w}_{d}(\infty)}_{0}\big{(}\mathcal{W}^{d}_{i}(\bullet)\big{)}$.
By Lemma 4.25, every element in
$H^{\mathcal{L}^{w}_{d}(\infty)}_{0}\big{(}\mathcal{W}^{d}_{i}(\bullet)\big{)}$
can be written as a sum of products of wheels on at least $d+1$ disks, and
averaged-filters.

When $2d+1\leq w$, the least efficient of these generators in terms of
homological degree to the number of disks are the wheels on $d+1$ disks. When
$|\bullet|=(d+1)i$, the quotient
$H^{\mathcal{L}^{w}_{d-1}(\infty)}_{0}\big{(}\mathcal{W}^{d}_{i}(\bullet)\big{)}\neq
0$, as the product of $i$ wheels on $d+1$ disks is nontrivial. For
$|\bullet|>(d+1)i$, any element in
$H_{\frac{(d)|\bullet|+i}{d}}\big{(}\text{conf}(\bullet,w);\mathbb{Q}\big{)}$
can be written as a sum elements, each of which includes a wheel on fewer than
$d+1$ disks not inside an averaged-filter. Such products are trivial in
$H^{\mathcal{L}^{w}_{d}(\infty)}_{0}\big{(}\mathcal{W}^{d}_{i}(\bullet)\big{)}$
by Lemma 4.25. Therefore, $\mathcal{W}^{d}_{i}(\bullet)$ is generated in
degree $\leq(d+1)i$ as an $\mathcal{L}^{w}_{d}(\infty)$-module. Since each
term of $\mathcal{W}^{d}_{i}(\bullet)$ is finite dimensional, it follows that
$\mathcal{W}^{d}_{i}(\bullet)$ is finitely generated as an
$\mathcal{L}^{w}_{d}(\infty)$-module.

If $2d=w$, the least efficient of the generators in terms of homological
degree to the number of disks are the proper averaged-filters on $w+1$ disks.
When $|\bullet|=(d+1)i+d$, the quotient
$H^{\mathcal{L}^{w}_{d}(\infty)}_{0}\big{(}\mathcal{W}^{d}_{i}(\bullet)\big{)}\neq
0$, as the product of $i$ averaged-filters each on $w+1$ disks is nontrivial.
For $|\bullet|>(d+1)i+d$, it follows that any product of elements in
$H_{\frac{(d-1)|\bullet|+i}{d}}\big{(}\text{conf}(\bullet,w);\mathbb{Q}\big{)}$
can be written as a sum of elements each of which includes a wheel on fewer
than $d+1$ disks. Such products are trivial in
$H^{\mathcal{L}^{w}_{d}(\infty)}_{0}\big{(}\mathcal{W}^{d}_{i}(\bullet)\big{)}$
by Lemma 4.25. It follows that $\mathcal{W}^{d}_{i}(\bullet)$ is generated in
degree $\leq(d+1)i+d$ as an $\mathcal{L}^{w}_{d}(\infty)$-module. Since each
term of $\mathcal{W}^{d}_{i}(\bullet)$ is finite dimensional,
$\mathcal{W}^{d}_{i}(\bullet)$ is finitely generated as an
$\mathcal{L}^{w}_{d}(\infty)$-module. ∎

We use Proposition 4.26 to prove the following theorem which shows that the
ordered configuration space of open unit diameter disks in the infinite strip
of width $w$ exhibits reasonable notions of $1^{\text{st}}-$ through
$\big{\lfloor}\frac{w}{2}\big{\rfloor}^{\text{th}}$-order representation
stability.

######  Theorem 4.27.

Let

$\mathcal{W}^{d}_{i}(\bullet):=H^{\mathcal{L}_{d-1}(\infty)}_{0}\Bigg{(}\cdots\bigg{(}H^{\mathcal{L}_{1}(\infty)}_{0}\Big{(}H_{\frac{(d-1)|\bullet|+i}{d}}\big{(}\text{conf}(\bullet,w);\mathbb{Q}\big{)}\Big{)}\bigg{)}\cdots\Bigg{)}.$

For $1\leq d\leq\big{\lfloor}\frac{w}{2}\big{\rfloor}$,

$\mathcal{W}^{d}_{i}(\bullet)$

is a finitely generated $\mathcal{L}_{d}(b+1)$-module, where
$b=\big{\lfloor}\frac{di}{w-d}\big{\rfloor}$, generated in degree $(d+1)i$ for
$2d+1\leq w$, and degree $(d+1)i+d$ for $2d=w$.

###### Proof.

By Proposition 4.26, for $1\leq d\leq\frac{w}{2}$,

$\mathcal{W}^{d}_{i}(\bullet):=H^{\mathcal{L}^{w}_{d-1}(\infty)}_{0}\Bigg{(}\cdots\bigg{(}H^{\mathcal{L}^{w}_{1}(\infty)}_{0}\Big{(}H_{\frac{(d-1)|\bullet|+i}{d}}\big{(}\text{conf}(\bullet,w);\mathbb{Q}\big{)}\Big{)}\bigg{)}\cdots\Bigg{)}$

is a finitely generated $\mathcal{L}^{w}_{d}(\infty)$-module generated in
degree $(d+1)i$ for $w\geq 2d+1$, and degree $(d+1)i+d$ for $w=2d$.

We count the maximum number of barriers in an element in
$\mathcal{W}^{d}_{i}(\bullet)$ as this will allow us to replace
$\mathcal{L}^{w}_{d}(\infty)$ with the finitely generated twisted algebra
$\mathcal{L}^{w}_{d}(b+1)$ without loss of information.

For $2d+1\leq w$, the module $\mathcal{W}^{d}_{i}(\bullet)$ is generated in
degree at most $(d+1)i$. This occurs in homological degree

$\frac{(d-1)(d+1)i+i}{d}=di.$

By Lemma 4.25, every element in
$H^{\mathcal{L}^{w}_{d}(\infty)}_{0}\big{(}\mathcal{W}^{d}_{i}(\bullet)\big{)}$
can be written as a product of wheels on at least $d+1$ disks, and averaged-
filters. The smallest of these generators that are also rational $d$-barriers
are the wheels on $w+1-d$ disks, which have homological degree $w-d$. We can
have at most $\big{\lfloor}\frac{di}{w-d}\big{\rfloor}$ rational $d$-barriers
in an element of $\mathcal{W}^{d}_{i}(\bullet)$. Since multiplication is well
defined by Proposition 4.21, we only need the first
$b+1:=\big{\lfloor}\frac{di}{w-d}\big{\rfloor}+1$ of the copies of
$H_{d-1}\big{(}\text{conf}(d,w);\mathbb{Q}\big{)}$ in
$\mathcal{L}^{w}_{d}(\infty)$, as multiplication by the other terms is equal
to multiplication by the $(b+1)^{\text{th}}$ copy of
$H_{d-1}\big{(}\text{conf}(d,w);\mathbb{Q}\big{)}$. Therefore, we can restrict
$\mathcal{L}^{w}_{d}(\infty)$ to $\mathcal{L}^{w}_{d}(b+1)$, and not lose
information on the structure of $\mathcal{W}^{d}_{i}(\bullet)$ as an
$\mathcal{L}^{w}_{d}(\infty)$-module. It follows from Proposition 4.26, that
for $w\geq 2d+1$, $\mathcal{W}^{d}_{i}(\bullet)$ is a finitely generated
$\mathcal{L}^{w}_{d}(b+1)$-module generated in degree $(d+1)i$.

For $2d=w$, the module $\mathcal{W}^{d}_{i}(\bullet)$ is generated in degree
at most $(d+1)i+d$. This occurs in homological degree

$\frac{(d-1)\big{(}(d+1)i+d\big{)}+i}{d}=\frac{d^{2}i-i+d^{2}-d+i}{d}=di+d-1.$

By Lemma 4.25, every element in
$H^{\mathcal{L}^{w}_{d}(\infty)}_{0}\big{(}\mathcal{W}^{d}_{i}(\bullet)\big{)}$
can be written as a product of wheels on at least $d+1$ disks, and averaged-
filters. The smallest of these generators that are also rational $d$-barriers
in terms of the homological degree are the wheels on $w+1-d$ disks, which have
homological degree $w-d$. Therefore, since $2d=w$, we can have at most

$\bigg{\lfloor}\frac{di+d-1}{w-d}\bigg{\rfloor}=\bigg{\lfloor}\frac{di+d-1}{d}\bigg{\rfloor}=\bigg{\lfloor}\frac{di}{d}+\frac{d-i}{d}\bigg{\rfloor}=\bigg{\lfloor}\frac{di}{w-d}+\frac{d-i}{d}\bigg{\rfloor}=\bigg{\lfloor}\frac{di}{w-d}\bigg{\rfloor}$

of these rational $d$-barriers. Since multiplication is well defined by
Proposition 4.21, we only need the first
$b+1:=\big{\lfloor}\frac{di}{w-d}\big{\rfloor}+1$ of the copies of
$H_{d-1}\big{(}\text{conf}(d,w);\mathbb{Q}\big{)}$ in
$\mathcal{L}^{w}_{d}(\infty)$, as multiplication by the other generators is
equal to multiplication by the generator of the $(b+1)^{\text{th}}$ copy of
$H_{d-1}\big{(}\text{conf}(d,w);\mathbb{Q}\big{)}$. Therefore, we can restrict
$\mathcal{L}^{w}_{d}(\infty)$ to $\mathcal{L}^{w}_{d}(b+1)$ and not lose
information on the structure of $\mathcal{W}^{d}_{i}(\bullet)$ as an
$\mathcal{L}^{w}_{d}(\infty)$-module. It follows from Proposition 4.26, that
for $w\geq 2d+1$, the $\mathcal{L}^{w}_{d}(b+1)$-module
$\mathcal{W}^{d}_{i}(\bullet)$ is finitely generated in degree $(d+1)i$. ∎

Our proof of Theorem 4.27 shows that for $i$ and $|\bullet|$ finite, there
exist integer valued functions $b_{m}(i,|\bullet|)$ such that one can replace
the $H^{\mathcal{L}^{w}_{m}(\infty)}_{0}$ in the definition of
$\mathcal{W}^{d}_{i}(\bullet)$ with
$H^{\mathcal{L}^{w}_{m}\big{(}b_{m}(i,|\bullet|)+1\big{)}}_{0}$, as all but
the first $b_{m}(i,|\bullet|)$ of the copies of
$H_{(m-1)}\big{(}\text{conf}(m,w);\mathbb{Q}\big{)}$ in
$\mathcal{L}^{w}_{m}(\infty)$ will prove to be redundant. Thus, we have shown
that the ordered configuration space of unit diameter disks in the infinite
strip of width $w$ exhibits notions of $1^{\text{st}}$ through
$\lfloor\frac{w}{2}\rfloor^{\text{th}}$-order representation stability.

In the spirit of [CEFN14, CEF15, Ram17, MW19], Theorem 4.27 suggests that if
we have a decomposition of $H_{d-1}\big{(}\text{conf}(d,w);\mathbb{Q}\big{)}$
into irreducible representations and we know how to decompose
$\mathcal{W}^{d}_{i}(n)$ into irreducible $S_{m}$-representations for all
$m\leq(d+1)i$ (or $m\leq(d+1)i+d$ for $w=2$), then we have upper bounds on the
multiplicities of the irreducible $S_{n}$-representations in a decomposition
of $\mathcal{W}^{d}_{i}(n)$ for $n>(d+1)i$ (or $n>(d+1)i+d$ for $w=2$).

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