# Primes in the Chebotarev density theorem
for all number fields
(With an Appendix by Andrew Fiori)

Habiba Kadiri  Department of Mathematics and Computer Science
University of Lethbridge
4401 University Drive
Lethbridge, Alberta
T1K 3M4 Canada<EMAIL_ADDRESS><EMAIL_ADDRESS>and  Peng-Jie
Wong  National Center for Theoretical Sciences
No. 1, Sec. 4, Roosevelt Rd., Taipei City, Taiwan<EMAIL_ADDRESS>
###### Abstract.

We establish an explicit bound for the least prime occurring in the Chebotarev
density theorem without any restriction. Let $L/K$ be any Galois extension of
number fields such that $L\not={\mathbb{Q}}$, and let $C$ be a conjugacy class
in the Galois group of $L/K$. We show that there exists an unramified prime
$\mathfrak{p}$ of $K$ such that $\sigma_{\mathfrak{p}}=C$ and
$N\mathfrak{p}\leq d_{L}^{B}$ with $B=310$. This improves the value
$B=12\,577$ as proven by Ahn and Kwon. In comparison to previous works on the
subject, we modify the weights to detect the least prime, and we use a new
version of Turán’s power sum method which gives a stronger Deuring-Heilbronn
(zero-repulsion) phenomenon. In addition, we refine the analysis of how the
location of the potential exceptional zero for $\zeta_{L}(s)$ affects the
final result. We also use Fiori’s numerical verification for $L$ up to a
certain discriminant height. Finally, we provide a lower bound for the number
of unramified primes $\mathfrak{p}$ of $K$ such that
$\sigma_{\mathfrak{p}}=C$.

###### Key words and phrases:

Deuring-Heilbronn phenomenon, Chebotarev density theorem, the least prime

###### 2010 Mathematics Subject Classification:

Primary 11R44, 11R42; Secondary 11Y35

This research was partially supported by the NSERC Discovery grants of H.K.
P.J.W. is currently an NCTS postdoctoral fellow; he was supported by a PIMS
postdoctoral fellowship and the University of Lethbridge during part of this
research.

##  1\. Introduction

Let $L/K$ be a Galois extension of number fields with Galois group $G$, and
let $C$ be a conjugacy class in $G$. The celebrated Chebotarev density theorem
asserts that

$\\#\\{\mathfrak{p}\subset\mathcal{O}_{K}\ |\ N\mathfrak{p}\leq x,\
\text{$\mathfrak{p}$ is an unramified prime with
$\sigma_{\mathfrak{p}}=C$}\\}\sim\frac{|C|}{|G|}\operatorname{Li}(x)$

as $x\to\infty$, where $\operatorname{Li}(x)$ is the usual logarithmic
integral, $\mathcal{O}_{K}$ is the ring of integers of $K$,
$N=N_{K/\mathbb{Q}}$ is the absolute norm of $K$, and $\sigma_{\mathfrak{p}}$
denotes the Artin symbol at $\mathfrak{p}$.

As the Chebotarev density theorem generalises Dirichlet’s theorem on primes in
arithmetic progressions, a natural question of finding the least (unramified)
prime $\mathfrak{p}$ with $\sigma_{\mathfrak{p}}=C$ then arises from Linnik’s
famous theorem on the least prime in an arithmetic progression. Under the
generalised Riemann hypothesis for the Dedekind zeta function $\zeta_{L}(s)$
of $L$, Lagarias and Odlyzko [LaOd77] showed that $N\mathfrak{p}\ll(\log
d_{L})^{2}$, where $d_{L}$ denotes the absolute discriminant of $L$ (cf.
[BaSo96]).

In [LaMoOd79], Lagarias, Montgomery, and Odlyzko proved, unconditionally, that
if $L\neq\mathbb{Q}$, then there is a constant $B>0$ such that there is an
unramified prime $\mathfrak{p}$ of $K$ with $\sigma_{\mathfrak{p}}=C$ and
$N\mathfrak{p}\leq d_{L}^{B}$. Recently, Zaman [Zam17] established that $B=40$
is valid when $d_{L}$ is sufficiently large. This is improved by Ng and the
authors in [KaNgWo19], who showed that $B=16$ is admissible for sufficiently
large $d_{L}$. Also, Ahn and Kwon [AhKw19-1] showed that $B=12\,577$ is valid
for _all_ number fields.

The main result of this article is the following theorem.

######  Theorem 1.

Let $L/K$ be a Galois extension of number fields with Galois group $G$, and
let $C$ be a conjugacy class in $G$. If $L\neq\mathbb{Q}$, then there exists
an unramified prime $\mathfrak{p}$ of $K$, of degree one, such that
$\sigma_{\mathfrak{p}}=C$ and $N\mathfrak{p}\leq d_{L}^{B}$ with $B=310$.

Tables 2 and 3 provide more refined results by degree and discriminant. For
the list of values $(n_{0},d_{0})$, Table 2 and respectively Table 3 list
admissible bounds $B$ for all fields of degree $n_{L}=n_{0}$ and with absolute
discriminant $d_{L}>d_{0}$ and $d_{L}\leq d_{0}$, respectively.

Note that without using the numerical verifications listed in Table 3, we can
still prove that $B=620$ is an admissible value for all fields with $n_{L}\geq
2$ and $d_{L}\geq 3$. The reader will find the proof in Section 3 and in
Fiori’s Appendix A.

Our major improvement comes from a stronger zero-repulsion theorem than the
one established in [AhKw19-1, Theorem 7.3]. Recall that there exist absolute
constants $A,A^{\prime}>0$ such that $\zeta_{L}(s)$ has no zeros in the region

${\mathfrak{Re}}(s)\geq 1-\frac{1}{A\log
d_{L}+A^{\prime}n_{L}\log(|{\mathfrak{Im}}(s)|+2)},$

with the exception of at most one real zero $\beta_{1}$, where $n_{L}$ is the
degree of $L$. It has been proven in [Kad12, Theorem 1.1] that for
$|{\mathfrak{Im}}(s)|\leq 1$, $A=12.74$, and $A^{\prime}=0$ assuming $d_{L}$
is sufficiently large. Recently, Lee improved this to $A=12.44$ in [Lee,
Theorem 2]. Also, this was made explicit in [AhKw19-1, Proposition 6.1] with
$A=A^{\prime}=29.57$. It is known that if such an exceptional real zero
$\beta_{1}$ exists, then other zeros of $\zeta_{L}(s)$ will be “repulsed”
further left. This is often called the Deuring-Heilbronn phenomenon. We give
here some versions of [KaNgWo19, Theorems 1.2 and 1.3] which are valid for all
number fields.

######  Theorem 2.

Let $L\neq\mathbb{Q}$ be a number field of degree $n_{L}$ and with absolute
discriminant $d_{L}$. Assume $\zeta_{L}(s)$ admits an exceptional real zero
$\beta_{1}$. Let $\beta^{\prime}+it$ be another (non-trivial) zero of
$\zeta_{L}(s)$, and let $\tau=|t|+2$. Then, for any $\eta\in(0,1]$, there
exist $c_{1}$ and $c_{2}$ such that either $\beta^{\prime}\leq 1-\eta$ or

$\beta^{\prime}\leq
1-c_{2}\frac{\log\Big{(}\frac{c_{1}}{(1-\beta_{1})\log(d_{L}\tau^{n_{L}})}\Big{)}}{\log(d_{L}\tau^{n_{L}})}.$
(1.1)

In addition, if $|t|\leq 1$, then there exist $c_{1}^{\prime}$ and
$c_{2}^{\prime}$ such that either $\beta^{\prime}\leq 1-\eta$ or

$\beta^{\prime}\leq
1-c_{2}^{\prime}\frac{\log\Big{(}\frac{c_{1}^{\prime}}{(1-\beta_{1})(\log
d_{L})}\Big{)}}{\log d_{L}}.$ (1.2)

Here $(c_{1},c_{2})$ and $(c_{1}^{\prime},c_{2}^{\prime})$ depend on $\eta$,
and are respectively defined in (2.24) and (2.28). For $\eta=1$, numerical
results are listed in Table 1.

In addition, we investigate the case of real zeros since as this case offers a
stronger zero-repulsion phenomenon, as illustrated in the next theorem.

######  Theorem 3.

Assume $\zeta_{L}(s)$ admits an exceptional real zero $\beta_{1}$. Let
$\beta^{\prime}$ be another real zero of $\zeta_{L}(s)$. Then, for any
$\eta\in(0,1]$, there exist $c_{1}^{\prime\prime}$ and $c_{2}^{\prime\prime}$
such that either $\beta^{\prime}\leq 1-\eta$ or

$\beta^{\prime}\leq
1-c_{2}^{\prime\prime}\frac{\log\Big{(}\frac{c_{1}^{\prime\prime}}{(1-\beta_{1})\log
d_{L}}\Big{)}}{\log d_{L}},$ (1.3)

where $(c^{\prime\prime}_{1},c^{\prime\prime}_{2})$, depend on $\eta$, are
defined in (2.30). For $\eta=1$, numerical results are listed in Table 1.

For instance, we establish that for $\eta=1$, that
$c_{2}=0.04485=\frac{1}{22.30}$, $c_{2}^{\prime}=0.05=\frac{1}{20}$, and
$c^{\prime\prime}_{2}=0.1008=\frac{1}{9.921}$ are admissible. This enlarges
the region described in [AhKw19-1, Theorem 7.3(2)], where $c_{2}=\frac{1}{77}$
was obtained.111The labelling for our $(c_{1},c_{2})$ was $(c_{7},c_{8})$ in
[AhKw19-1, Theorem 7.3(2)]. In addition this extends and improves [Zam17,
Theorem 1.2] where Zaman proved $c_{2}^{\prime}=\frac{1}{35.8}$ for $d_{L}$
sufficiently large. In [KaNgWo19], the repulsion constant was improved to
$\frac{1}{14.144}$, for $d_{L}$ sufficiently large.

We refer the reader to Section 2.2 for a proof of Theorems 2 and 3. The key
tool resides in Theorem 7 where we improve lower bounds for some modified
Turán power sums.

In Section 2.3 we deduce a bound for the possible exceptional zero $\beta_{1}$
of $\zeta_{L}(s)$.

######  Theorem 4.

Suppose $L\neq\mathbb{Q}$. If $\zeta_{L}(s)$ admits a real zero $\beta_{1}$,
then

$1-\beta_{1}\geq d_{L}^{-c_{3}}\ \text{with}\ c_{3}=11.7.$

This improves [AhKw19-1, Corollary 7.4] where $c_{3}=114.72...$ was proven
admissible. We note that for $d_{L}$ sufficiently large, the bound above was
established with $c_{3}=16.6$ in [Zam17, Corollary 1.4] and $c_{3}=7.072$ in
[KaNgWo19, Corollary 1.3.1].

We let $\tilde{\pi}_{C}(x)$ denote the number of degree-one unramified primes
$\mathfrak{p}$ of $K$ such that $N\mathfrak{p}\leq x$ and
$\sigma_{\mathfrak{p}}=C$. In Section 4 we apply Theorem 4 to obtain a lower
bound for $\tilde{\pi}_{C}(x)$.

######  Theorem 5.

Let $L/K$ be a Galois extension of number fields with Galois group $G$, and
let $C$ be a conjugacy class in $G$. If $L\neq\mathbb{Q}$, then for
$x\geq\exp(d_{L}^{c_{3}})$, we have

$\tilde{\pi}_{C}(x)\geq m\frac{|C|}{|G|}\frac{x}{\log x},$

where $c_{3}=11.7$ and $m=0.4899$.

This improves [AhKw19-2, Theorem 3] where Ahn and Kwon obtained instead
$c_{3}=114.72...$ and $m=0.353$ for all $L\neq{\mathbb{Q}}$.

We also note that Thorner and Zaman [ThZa17, Theorem 3.1] showed that there
are absolute constants $\kappa_{2}$ and $\kappa_{3}$ such that

$\tilde{\pi}_{C}(x)\gg\frac{1}{d_{L}^{\kappa_{2}}}\frac{|C|}{|G|}\operatorname{Li}(x)$

for $x\geq d_{L}^{\kappa_{3}}$ and $d_{L}$ sufficiently large.

While our range for $x$ is more restricted, we are able to obtain a lower
bound independent of $L$.

### Notation

Throughout this article, we let $L$ be a number field of degree $n_{L}\geq
n_{0}\geq 2$ and with absolute discriminant $d_{L}\geq d_{0}\geq 3$.

We denote

${\mathscr{L}}=\log d_{L}\ \text{and}\ {\mathscr{L}}_{0}=\log d_{0}.$ (1.4)

Together with Minkowski’s bound, we consider

$n_{0}\leq n_{L}\leq\frac{2\log d_{L}}{\log 3},\ d_{L}\geq d_{0},\ \text{and}\
\log d_{L}\geq{\mathscr{L}}_{0}.$ (1.5)

Let $\tau=|t|+2$. We shall define

$\delta=\delta_{L}(\tau)=\frac{\log(\tau^{n_{L}})}{\log d_{L}},$ (1.6)

and use various bounds for $\delta$ depending on cases. For $T_{0}\geq 0$, we
define

$\Delta_{0}(T_{0})=\mathscr{Q}_{0}\log(T_{0}+2),$ (1.7)

where

$\mathscr{Q}_{0}=\begin{cases}\frac{n_{L}}{{\mathscr{L}}_{0}}&\textrm{ if
$2\leq n_{L}\leq 20$;}\\\ \frac{1}{\log 10}&\textrm{ if $n_{L}\geq
21$.}\end{cases}$ (1.8)

Note that for $|t|\leq T_{0}$, we have

$0\leq\delta\leq\Delta_{0}(T_{0}).$ (1.9)

In Section 2, we prove in all generality the zero-repulsion results for all
$n_{L}\geq 2$.

In Section 3, we prove the least prime result for $(n_{0},d_{0})$ as listed in
Tables 2 and 3. In Appendix A, numerical verifications of the bound for the
least prime are done for $n_{L}=n_{0}$ with $2\leq n_{0}\leq 20$ and
$d_{L}\leq d_{0}$, as well as for $n_{L}\geq n_{0}\geq 21$ and $d_{L}\leq
d_{0}=10^{n_{L}}$.

##  2\. Repelling non-exceptional zeros further left

We recall [AhKw19-1, Proposition 6.1], which asserts that for any number field
$L$, there are no zeros for $\zeta_{L}(s)$ in the region

${\mathfrak{Re}}(s)\geq
1-\frac{1}{29.57\left({\mathscr{L}}+n_{L}\log(|{\mathfrak{Im}}(s)|+2)\right)},$
(2.1)

with the exception of at most one real zero $\beta_{1}$. If the exceptional
zero $\beta_{1}$ exists, then it has to be real and simple. In addition, by
[Kad12] (for $d_{L}$ sufficiently large) and by [AhKw14, Theorem 1] (for all
$L\neq\mathbb{Q}$), we know that $\zeta_{L}(s)$ has at most one zero in the
region

$\beta>1-\frac{1}{R_{0}{\mathscr{L}}}\text{ and
}|\gamma|<\frac{1}{2{\mathscr{L}}},$ (2.2)

for any $R_{0}\geq 2$. In what follows we let $\beta_{1}$ denote the
exceptional zero of $\zeta_{L}(s)$ if it exists. We shall split our
considerations into several cases depending on the location (and existence) of
$\beta_{1}$.

###  2.1. Enlarging the region without non-exceptional zeros

We assume that there exists $R_{1}\leq R_{0}$ such that there is no non-
trivial zeros $\rho\neq\beta_{1}$ of $\zeta_{L}(s)$ in the region

$\beta>1-\frac{1}{R_{1}{\mathscr{L}}}\text{ and
}|\gamma|<\frac{1}{R_{1}{\mathscr{L}}}.$ (2.3)

We note that one can always take $R_{1}=2$, but show here that we can take
1.24 and 1.7 as admissible values for $R_{1}$ for certain instances:

######  Proposition 6.

Let $L$ be a number field such that $L\not={\mathbb{Q}}$, and let $R_{0}\geq
2.$ If the exceptional zero $\beta_{1}$ of $\zeta_{L}(s)$ presents in
$(1-\frac{1}{R_{0}{\mathscr{L}}},1)$, then there is no other zeros of
$\zeta_{L}(s)$ in the region

$\beta>1-\frac{1}{1.7{\mathscr{L}}}\text{ and
}|\gamma|<\frac{1}{1.7{\mathscr{L}}}.$ (2.4)

Moreover, if $R_{0}\geq 3.5$, then $1.7$ in (2.4) can be improved to $1.24$.

###### Proof of Proposition 6.

Suppose that the exceptional zero $\beta_{1}$ presents in
$(1-\frac{1}{R_{0}{\mathscr{L}}},1)$. Let $\beta_{2}+i\gamma_{2}$ be another
non-trivial zero of $\zeta_{L}(s)$ such that

$\beta_{2}>1-\frac{1}{R{\mathscr{L}}}\text{ and
}|\gamma_{2}|<\frac{1}{R{\mathscr{L}}}.$

Following [AhKw14, Sec. 2], we assume $R\geq 1.24$ and set
$\sigma=1+\frac{1}{r{\mathscr{L}}},$ with $1<\sigma\leq 6.2$, and
$\frac{1}{5.2\log 3}\leq r\leq R$. As argued in [AhKw14, pp. 438–439], one has

$\Big{(}\frac{1}{2}\Big{(}1-\frac{1}{\sqrt{5}}\Big{)}+r\Big{)}{\mathscr{L}}\geq\frac{\sigma-\beta_{1}}{(\sigma-\beta_{1})^{2}}+\frac{\sigma-\beta_{2}}{(\sigma-\beta_{2})^{2}+\gamma_{2}^{2}}=\frac{1}{\sigma-\beta_{1}}+\frac{\sigma-\beta_{2}}{(\sigma-\beta_{2})^{2}+\gamma_{2}^{2}},$

and thus

$\frac{1}{2}\Big{(}1-\frac{1}{\sqrt{5}}\Big{)}+r\geq\frac{rR_{0}}{R_{0}+r}+\frac{rR(R+r)}{(R+r)^{2}+r^{2}}.$
(2.5)

By a numerical calculation, if $R_{0}\geq 2$ and $r=0.6$, it can be checked
that (2.5) fails for $R\geq 1.7$. Also, if $R_{0}\geq 3.5$ and $r=0.6$, (2.5)
fails for $R\geq 1.24$. This concludes the proof. ∎

###  2.2. Deuring-Heilbronn phenomenon for all number fields

In this section, we prove Theorems 2 and 3, that is quantitative versions of
the Deuring-Heilbronn phenomenon for all number fields. To do so, we employ
the Turán’s power sum method as introduced by Lagarias, Montgomery, and
Odlyzko in [LaMoOd79] (see also Montgomery’s [Mon94, Theorem 11]). We note
that Ahn and Kwon employ [LaMoOd79, Theorem 4.2] to prove their version
[AhKw19-1, Theorem 7.3] of the Deuring-Heilbronn phenomenon for all number
fields. The reader can also find a refinement of [LaMoOd79, Theorem 4.2] in
Zaman’s [Zam17, Theorem 2.3]. In [KaNgWo19, Theorem 2.2], the authors,
together with Ng, appeal to Harnack’s inequality to bring some improvements to
these results. The following is the case $s_{j}=\sum_{n\geq 1}b_{n}z_{n}^{j}$
with $b_{n}=1$ for all $n$.

######  Theorem 7.

Let $\varepsilon>0$ and $z_{1}\neq 0$. Assume that for all $n\geq 1$, $z_{n}$
is a complex number satisfying $|z_{n}|\leq|z_{1}|$. For any $j\in\mathbb{N}$,
set $s_{j}=\sum_{n\geq 1}z_{n}^{j}$ and $M=\sum_{n\geq
1}\frac{|z_{n}|}{|z_{1}|+|z_{n}|}$. Then there exists $j_{0}$ with $1\leq
j_{0}\leq(8+\varepsilon)M$ such that

${\mathfrak{Re}}(s_{j_{0}})\geq\frac{\varepsilon}{4(8+\varepsilon)}|z_{1}|^{j_{0}}.$

We propose here a bound for ${\mathfrak{Re}}\frac{\Gamma^{\prime}}{\Gamma}$
which is the last tool we need to prove Theorem 2. It improves [AhKw19-1,
Lemma 5.3] and will allow for sharper estimates for
${\mathfrak{Re}}\frac{\gamma_{L}^{\prime}}{\gamma_{L}}$ than [AhKw19-1, Lemma
5.4], where $\gamma_{L}$ is the associated gamma factor of $\zeta_{L}(s)$.

######  Lemma 8.

Let $T_{0}\geq 0$. For $s=\sigma+it$ with $\sigma>0$ and $|t|\leq T_{0}$, we
define

$g(\sigma,T_{0})=\max_{|t|\leq
T_{0}}\Big{(}\log\Big{(}\frac{\sqrt{\sigma^{2}+t^{2}}}{|t|+2}\Big{)}-\frac{\sigma}{\sigma^{2}+t^{2}}\Big{)}+\frac{1}{3\sigma^{2}}-\log
2,$ (2.6)

Then

${\mathfrak{Re}}\frac{\Gamma^{\prime}}{\Gamma}\Big{(}\frac{s}{2}\Big{)}\leq\log(|t|+2)+g(\sigma,T_{0}).$
(2.7)

###### Proof.

According to [AhKw19-1, p. 1429],

${\mathfrak{Re}}\frac{\Gamma^{\prime}}{\Gamma}(s)\leq\log|s|-\frac{\sigma}{2(\sigma^{2}+t^{2})}+\frac{1}{12\sigma^{2}}.$
(2.8)

Thus,

${\mathfrak{Re}}\frac{\Gamma^{\prime}}{\Gamma}\Big{(}\frac{s}{2}\Big{)}\leq\log(|t|+2)+\max_{|t|\leq
T_{0}}\Big{(}\log\Big{(}\frac{\sqrt{\sigma^{2}+t^{2}}}{|t|+2}\Big{)}-\frac{\sigma}{\sigma^{2}+t^{2}}\Big{)}+\frac{1}{3\sigma^{2}}-\log
2$

from which (2.7) follows. ∎

Now, we are in a position to prove Theorems 2 and 3. For both proofs, we are
assuming (1.5) for $n_{L}$ and $d_{L}$, and we introduce
$\varepsilon>0,\sigma\geq 2$, and $\eta\in(0,1]$.

###### Proof of Theorem 2.

We denote $\beta_{1}$ the exceptional real zero that we assume $\zeta_{L}(s)$
possesses. We let $\mathcal{S}$ be the set of all non-trivial zeros of
$\zeta_{L}(s)$. Let $\eta\in(0,1]$ and $T_{0}\geq 0$. We also denote
$\beta^{\prime}+it$ a non-trivial zero of $\zeta_{L}(s)$ such that
$1-\eta\leq\beta^{\prime}<1$ and $|t|\leq T_{0}$. We shall let $\sigma\geq 2$
and let $z_{n}$ run over $(\sigma-\rho)^{-2}$ and $(\sigma+it-\rho)^{-2}$ for
all $\rho\in\mathcal{S}\backslash\\{\beta_{1}\\}$ in order that $|z_{n}|$
decreases (so $|z_{1}|\geq|z_{2}|\geq\cdots$). Thus,
$|z_{1}|\geq\frac{1}{(\sigma-\beta^{\prime})^{2}}$, and by the inequality
$\log(1+x)\leq x$, we have

$|z_{1}|\geq\frac{1}{(\sigma-1)^{2}}\frac{(\sigma-1)^{2}}{(\sigma-\beta^{\prime})^{2}}\geq\frac{1}{(\sigma-1)^{2}}\exp\Big{(}-2\log\Big{(}\frac{\sigma-\beta^{\prime}}{\sigma-1}\Big{)}\Big{)}\geq\frac{1}{(\sigma-1)^{2}}\exp\Big{(}-2\frac{1-\beta^{\prime}}{\sigma-1}\Big{)}.$
(2.9)

From [AhKw19-1, Eq. (7.2) and Lemma 7.1], it follows that

${\mathfrak{Re}}\sum_{n\geq
1}z_{n}^{j_{0}}\leq\frac{4j_{0}(1-\beta_{1})}{(\sigma-1)^{2j_{0}+1}}.$ (2.10)

Applying Theorem 7 for $\varepsilon>0$, there exists $j_{0}$ with $1\leq
j_{0}\leq(8+\varepsilon)M$ such that

${\mathfrak{Re}}\sum_{n\geq
1}z_{n}^{j_{0}}\geq\frac{\varepsilon}{4(8+\varepsilon)}|z_{1}|^{j_{0}}.$
(2.11)

Combining (2.9), (2.10), (2.11), and the fact $j_{0}\leq(8+\varepsilon)M$
gives

$1-\beta^{\prime}\geq\frac{\sigma-1}{2(8+\varepsilon)M}\log\Big{(}\frac{\varepsilon(\sigma-1)}{16M(8+\varepsilon)^{2}(1-\beta_{1})}\Big{)}.$
(2.12)

We finalise by appealing to [KaNgWo19, p. 2294] to establish an upper bound
for $M$:

$M\leq\mathcal{A}S_{L}(d-1,t),$ (2.13)

where $\mathcal{A}$, $d$, and $S_{L}(d-1,t)$ are respectively defined in
[KaNgWo19, Eq. (2.6), (2.11), and (2.12)]:

$\displaystyle\mathcal{A}=\mathcal{A}(\sigma,\eta)=(\sigma-1+\eta)^{2},$
(2.14) $\displaystyle
d=d(\sigma,\eta)=\sqrt{\sigma^{2}+\mathcal{A}}=\sqrt{2\sigma^{2}+(1-\eta)^{2}-2\sigma(1-\eta)},$
(2.15) $\displaystyle
S_{L}(d-1,t)=\sum_{\rho}\Big{(}\frac{1}{|d-\rho|^{2}}+\frac{1}{|d+it-\rho|^{2}}\Big{)}.$
(2.16)

Also, [Zam17, Lemma 2.5 and Eq. (2.10)] provides a bound for $S_{L}$:

$S_{L}(d-1,t)\leq\frac{\log
d_{L}}{d-1}+\frac{G_{1}(d-1;|t|)}{d-1}r_{1}+\frac{G_{2}(d-1;|t|)}{d-1}(2r_{2})+\frac{2}{(d-1)^{2}}+\frac{2}{d(d-1)},$
(2.17)

where $r_{1}$ and $r_{2}$ are the numbers of real and complex places,
respectively, of $L$, and

$\begin{split}G_{1}(d-1;|t|)r_{1}+G_{2}(d-1;|t|)(2r_{2})=&\frac{r_{1}+r_{2}}{2}\frac{\Gamma^{\prime}}{\Gamma}\Big{(}\frac{d}{2}\Big{)}+\frac{r_{1}+r_{2}}{2}{\mathfrak{Re}}\frac{\Gamma^{\prime}}{\Gamma}\Big{(}\frac{d+i|t|}{2}\Big{)}\\\
&+\frac{r_{2}}{2}\frac{\Gamma^{\prime}}{\Gamma}\Big{(}\frac{d+1}{2}\Big{)}+\frac{r_{2}}{2}{\mathfrak{Re}}\frac{\Gamma^{\prime}}{\Gamma}\Big{(}\frac{d+1+i|t|}{2}\Big{)}-n_{L}\log\pi.\end{split}$
(2.18)

Together with the bounds on Gamma from Lemma 8 and the fact that
$\frac{\Gamma^{\prime}}{\Gamma}(x)$ increases with real values $x$, we obtain

$r_{1}G_{1}(d-1;|t|)+2r_{2}G_{2}(d-1;|t|)\leq\frac{\log(\tau^{n_{L}})}{2}+\frac{n_{L}}{2}\Big{(}\frac{\Gamma^{\prime}}{\Gamma}\Big{(}\frac{d+1}{2}\Big{)}+\max_{x=d,d+1}g(x,T_{0})-2\log\pi\Big{)}.$
(2.19)

We observe that, for $\delta$ defined in (1.6), whenever $\log
d_{L}\geq{\mathscr{L}}_{0}$,

$\displaystyle\log
d_{L}=\frac{1}{1+\delta}\log\Big{(}d_{L}\tau^{n_{L}}\Big{)},\quad\log(\tau^{n_{L}})=\frac{\delta}{1+\delta}\log(d_{L}\tau^{n_{L}}),$
$\displaystyle n_{L}\leq\frac{\delta}{1+\delta}\frac{1}{\log
2}\log(d_{L}\tau^{n_{L}}),\quad
1\leq\frac{1}{(1+\delta)}\frac{1}{{\mathscr{L}}_{0}}\log\Big{(}d_{L}\tau^{n_{L}}\Big{)}.$

We combine these with (2.19) so that (2.17) becomes

$S_{L}(d-1,t)\leq\frac{\log(d_{L}\tau^{n_{L}})}{d-1}\Big{(}\frac{b_{1}(d)+b_{2}(d,T_{0})\delta}{1+\delta}\Big{)},$
(2.20)

where we denote

$\displaystyle
b_{1}(d)=1+\frac{2}{{\mathscr{L}}_{0}(d-1)}+\frac{2}{{\mathscr{L}}_{0}d},$
(2.21) $\displaystyle b_{2}(d,T_{0})=\frac{1}{2}+\frac{1}{2\log
2}\max\Big{\\{}\frac{\Gamma^{\prime}}{\Gamma}\Big{(}\frac{d+1}{2}\Big{)}+\max\\{g(d,T_{0}),g(d+1,T_{0})\\}-2\log\pi,0\Big{\\}}.$
(2.22)

Now, setting

$\mathcal{B}=\mathcal{B}(d,T_{0})=\frac{1}{d-1}\max_{\delta\in[0,\Delta_{0}(T_{0})]}\Big{(}\frac{b_{1}(d)+b_{2}(d,T_{0})\delta}{1+\delta}\Big{)},$
(2.23)

we see that (2.13) becomes
$M\leq\mathcal{A}\mathcal{B}\log(d_{L}\tau^{n_{L}}).$ From (2.12) and (2.13),
it follows that

$1-\beta^{\prime}\geq\frac{\sigma-1}{2(8+\varepsilon)\mathcal{A}\mathcal{B}}\frac{\log\Big{(}\frac{\varepsilon(\sigma-1)}{16(8+\varepsilon)^{2}\mathcal{A}\mathcal{B}}\frac{1}{(1-\beta_{1})\log(d_{L}\tau^{n_{L}})}\Big{)}}{\log(d_{L}\tau^{n_{L}})}=c_{2}\frac{\log\Big{(}\frac{c_{1}}{(1-\beta_{1})\log(d_{L}\tau^{n_{L}})}\Big{)}}{\log(d_{L}\tau^{n_{L}})},$

with

$c_{1}=\frac{\varepsilon}{8(8+\varepsilon)}c_{2}\ \text{ and }\
c_{2}=\frac{\sigma-1}{2(8+\varepsilon)\mathcal{A}\mathcal{B}}.$ (2.24)

Now, taking $T_{0}=\infty$, we establish the first part of Theorems 2. We note
that one may bound $g(\sigma,\infty)$, trivially, by

$g(\sigma,\infty)\leq\frac{1}{2}\log\Big{(}\frac{\sigma^{2}}{4}+1\Big{)}+\frac{1}{3\sigma^{2}}-\log
2.$

We will use this bound to control $\mathcal{B}(d,\infty)$ and calculate
$c_{1}$ and $c_{2}$ for the theorem.

On the other hand, for $|t|\leq 1$ (so for $T_{0}=1$), we may directly use
(2.8) to deduce

$\max_{x=d,d+1}{\mathfrak{Re}}\frac{\Gamma^{\prime}}{\Gamma}\Big{(}\frac{x+it}{2}\Big{)}\leq\max_{x=d,d+1}\Big{(}\log\sqrt{x^{2}+1}-\frac{x}{x^{2}+1}+\frac{1}{3x^{2}}\Big{)}-\log
2.$ (2.25)

Hence, by (2.18), we have $G_{1}(d-1;t)r_{1}+G_{2}(d-1;t)(2r_{2})\leq
n_{L}\mathcal{H}(d),$ where we denote

$\mathcal{H}(d)=\frac{1}{2}\frac{\Gamma^{\prime}}{\Gamma}\Big{(}\frac{d+1}{2}\Big{)}+\frac{1}{2}\max_{x=d,d+1}\Big{(}\log\sqrt{x^{2}+1}-\frac{x}{x^{2}+1}+\frac{1}{3x^{2}}\Big{)}-\frac{\log
2+2\log\pi}{2},$ (2.26)

Thus, whenever $n_{L}\geq n_{0}$ and $\frac{n_{L}}{\log
d_{L}}\leq\mathscr{Q}_{0}$, we obtain

$\displaystyle S_{L}(d-1,t)\leq$ $\displaystyle\frac{1}{d-1}\Big{(}\log
d_{L}+n_{L}\mathcal{H}(d)+\frac{4d-2}{d(d-1)}\Big{)}$ $\displaystyle\leq$
$\displaystyle\frac{\log
d_{L}}{d-1}\Big{(}1+\mathscr{Q}_{0}\max\Big{\\{}\mathcal{H}(d)+\frac{4d-2}{d(d-1)n_{0}},0\Big{\\}}\Big{)},$

with $\mathscr{Q}_{0}$ as defined in (1.8). Now, setting

$\mathcal{B}^{\prime}=\mathcal{B}^{\prime}(d)=\frac{1+\mathscr{Q}_{0}\max\Big{\\{}0,\mathcal{H}(d)+\frac{4d-2}{d(d-1)n_{0}}\Big{\\}}}{d-1},$
(2.27)

(2.13) gives $M\leq\mathcal{A}\mathcal{B}^{\prime}\log d_{L}.$ Applying
(2.12), we arrive at $1-\beta^{\prime}\geq
c_{2}^{\prime}\frac{\log\Big{(}\frac{c_{1}^{\prime}}{(1-\beta_{1})\log
d_{L}}\Big{)}}{\log d_{L}},$ where

$c_{1}^{\prime}=c_{1}^{\prime}(\varepsilon,\sigma,\eta)=\frac{\varepsilon}{8(8+\varepsilon)}c_{2}^{\prime}\
\text{ and }\
c_{2}^{\prime}=c_{2}^{\prime}(\varepsilon,\sigma,\eta)=\frac{\sigma-1}{2(8+\varepsilon)\mathcal{A}\mathcal{B}^{\prime}}.$
(2.28)

This completes the proof of Theorem 2 ∎

We shall now prove the version of the Deuring-Heilbronn phenomenon that _only_
concerns the location of real zeros of $\zeta_{L}(s)$. We adapt the previous
proof to the case $t=0$.

###### Proof of Theorem 3.

The proof repeats most of the argument from the previous proof, so we shall
keep using the notation with $t=0$, and with the sequence $z_{n}$ runing over
$(\sigma-\rho)^{-2}$ for all the non-trivial zeros
$\rho\in\mathcal{S}\backslash\\{\beta_{1}\\}$, with $z_{n}$ satisfying
$|z_{1}|\geq|z_{2}|\geq\cdots$. It follows from [KaNgWo19, Proof of Lemma 2.7]
that $2M\leq\mathcal{A}S_{L}(d-1,0)$. We observe that

$\frac{S_{L}(d-1,0)}{2}\leq\frac{1}{d-1}\Big{(}\frac{\log
d_{L}}{2}+\frac{n_{L}}{2}\Big{(}\frac{\Gamma^{\prime}}{\Gamma}\Big{(}\frac{d+1}{2}\Big{)}-\log\pi\Big{)}+\frac{2d-1}{d(d-1)}\Big{)}.$

Thus, setting

$\mathcal{B}^{\prime\prime}=\mathcal{B}^{\prime\prime}(d)=\frac{1}{d-1}\Big{(}\frac{1}{2}+\mathscr{Q}_{0}\max\Big{\\{}0,\frac{1}{2}\frac{\Gamma^{\prime}}{\Gamma}\Big{(}\frac{d+1}{2}\Big{)}-\frac{\log\pi}{2}+\frac{2d-1}{d(d-1)n_{0}}\Big{\\}}\Big{)},$
(2.29)

we have
$M\leq\mathcal{A}\frac{S_{L}(d-1,0)}{2}\leq\mathcal{A}\mathcal{B}^{\prime\prime}(\log
d_{L})$ whenever $n_{L}\geq n_{0}$, and $\frac{n_{L}}{\log
d_{L}}\leq\mathscr{Q}_{0}$. Hence, arguing as before, we have
$1-\beta^{\prime}\geq
c_{2}^{\prime\prime}\frac{\log\Big{(}\frac{c_{1}^{\prime\prime}}{(1-\beta_{1})\log
d_{L}}\Big{)}}{\log d_{L}},$ where $c_{1}^{\prime\prime},c_{2}^{\prime\prime}$
depend on $(8+\varepsilon),\sigma,\eta$ and are given by

$c_{1}^{\prime\prime}=\frac{\varepsilon}{8(8+\varepsilon)}c_{2}^{\prime\prime}\
\text{ and }\
c_{2}^{\prime\prime}=\frac{\sigma-1}{2(8+\varepsilon)\mathcal{A}\mathcal{B}^{\prime\prime}}.$
(2.30)

This completes the proof of Theorem 3. ∎

###  2.3. Distance of the exceptional zero to 1-line

With Theorem 3 in hand, we shall prove Theorem 4 in this section.

###### Proof of Theorem 4.

We let $\eta\in(0,1]$ be a parameter to be chosen later. We consider $\beta$ a
non-exceptional real zero of $\zeta_{L}(s)$. We assume $\beta\geq 1-\eta$, and
we consider $c_{3}$ such that

$1-\beta_{1}=d_{L}^{-c_{3}}.$

By Theorem 3, whenever ${\mathscr{L}}\geq{\mathscr{L}}_{0}$, we have

$\eta\geq 1-\beta\geq c_{2}^{\prime\prime}\frac{\log
c_{1}^{\prime\prime}-\log(1-\beta_{1})-\log\log d_{L}}{\log d_{L}}\geq
c_{2}^{\prime\prime}\frac{\log
c_{1}^{\prime\prime}+c_{3}{\mathscr{L}}-\log{\mathscr{L}}}{{\mathscr{L}}},$

where $c_{1}^{\prime\prime}$ and $c_{2}^{\prime\prime}$ are defined in (2.30).
Thus,

$c_{3}\leq\frac{\eta}{c_{2}^{\prime\prime}}+\frac{\log{\mathscr{L}}-\log
c_{1}^{\prime\prime}}{{\mathscr{L}}}\leq\frac{\eta}{c_{2}^{\prime\prime}}+\frac{\log{\mathscr{L}}_{0}-\log
c_{1}^{\prime\prime}}{{\mathscr{L}}_{0}},$ (2.31)

which completes the proof of Theorem 4. ∎

######  Remark.

We note that if there are only two real zeros (i.e $\beta_{1}$ and
$1-\beta_{1}$), then we can only take the trivial bound $\beta=1-\beta_{1}>0$
(i.e. $\eta=1$). However, if there is a third real zero $\beta^{\prime}$, by
the symmetry of zeros of $\zeta_{L}(s)$, we may take $\beta=\beta^{\prime}\geq
1/2$ (so $\eta=1/2$ is admissible), which reduces the admissible value for
$c_{3}$ and, consequently, would push $\beta_{1}$ away from the 1-line a bit
further. In other words, while Deuring-Heilbronn describes how an exceptional
real zero would push further left other zeros in its vicinity, the other real
zeros also do the same to the exceptional zero. We note that this phenomenon
would hold for Dirichlet $L$-functions as well.

###  2.4. Numerical results

As Theorems 2, 3, and 4 are “trivial” if $\zeta_{L}(s)$ does not admit the
exceptional zero $\beta_{1}$, it is sufficient to calculate
$c_{1},c_{2},c_{1}^{\prime},c_{2}^{\prime},c_{1}^{\prime\prime},c_{2}^{\prime\prime},c_{3}$
for $L$ such that $\zeta_{L}(s)$ may admit the exceptional zero $\beta_{1}$.
To do so, we first summarise some cases that the non-existence of $\beta_{1}$
is known.

For $n_{L}=2$, we recall that there is a (quadratic) Dirichlet character
$\chi_{L}$ modulo $d_{L}$ such that $\zeta_{L}(s)=\zeta(s)L(s,\chi_{L}),$
where $\zeta(s)$ is the Riemann zeta function, and $L(s,\chi_{L})$ is the
Dirichlet $L$-function attached to $\chi_{d_{L}}$. It is known that
$\zeta(\sigma)$ is non-vanishing for $\sigma>0$. Moreover, Platt [Pla16]
showed that for any Dirichlet character $\chi$ modulo $q$, with $q\leq
400\,000$, $L(s,\chi)$, the Dirichlet $L$-function attached to $\chi$, has no
positive real zeros. Hence, for $n_{L}=2$, $\zeta_{L}(s)$ has no positive real
zeros if $d_{L}\leq 400\,000$.

For $n_{L}=3$ and $d_{L}\leq 239$, Tollis [Tol97] verified the generalised
Riemann hypothesis for $\zeta_{L}(s)$ to $|{\mathfrak{Im}}(s)|\leq 92$. Also,
for $n_{L}=4$ and $d_{L}\leq 320$, Tollis [Tol97] verified the generalised
Riemann hypothesis for $\zeta_{L}(s)$ to $|{\mathfrak{Im}}(s)|\leq 40$.
Consequently, $\zeta_{L}(s)$ has no positive real zeros whenever $n_{L}=3$ and
$d_{L}\leq 239$ or $n_{L}=4$ and $d_{L}\leq 320$. Hence, for $2\leq n_{L}\leq
4$, we only need to compute
$c_{1},c_{2},c_{1}^{\prime},c_{2}^{\prime},c_{1}^{\prime\prime},c_{2}^{\prime\prime},c_{3}$
for $L$ with $d_{L}\geq d_{0}$, where $d_{0}$ given in Table 1.

For fields $L$ of degree $n_{L}\geq 5$, instead of trying to confirm the non-
existence of the exceptional zero of $\zeta_{L}(s)$, we use the following
lower bounds $d_{0}$ of $d_{L}$ to calculate
$c_{1},c_{2},c_{1}^{\prime},c_{2}^{\prime},c_{1}^{\prime\prime},c_{2}^{\prime\prime},c_{3}$.
For $5\leq n_{L}\leq 8$, by [LMFDB], it can be checked that all the fields $L$
satisfy that $d_{L}\geq d_{0}$, where $d_{0}$ is given in Table 1. For $9\leq
n_{L}\leq 20$ As remarked by Fiori, by the lower bounds for the minimal root
discriminant in [DyD80], we know that $d_{L}\geq d_{0}$, where $d_{0}$ is
given in Table 1. In addition, for $n_{L}\geq 21$, by [DyD80], one has
$d_{L}>10^{n_{L}},$ and thus $d_{0}$ can be taken as $d_{0}=10^{n_{L}}$ for
these fields.

Table 1. Values for $(c_{1},c_{2},c_{1}^{\prime},c_{2}^{\prime},c_{1}^{\prime\prime},c_{2}^{\prime\prime},c_{3})$ as defined in Theorems 2, 3, and 4 $n_{0}$ | $d_{0}$ | $c_{1}$ | $c_{2}$ | $c_{1}^{\prime}$ | $c_{2}^{\prime}$ | $c_{1}^{\prime\prime}$ | $c_{2}^{\prime\prime}$ | $c_{3}$
---|---|---|---|---|---|---|---|---
$2$ | $400\,000$ | $5.174\cdot 10^{-5}$ | $0.04233$ | $7.904\cdot 10^{-5}$ | $0.06466$ | $1.581\cdot 10^{-4}$ | $0.1293$ | $8.608$
$3$ | $239$ | $1.276\cdot 10^{-4}$ | $0.05697$ | $1.133\cdot 10^{-4}$ | $0.05059$ | $2.275\cdot 10^{-4}$ | $0.1015$ | $11.69$
$4$ | $320$ | $1.047\cdot 10^{-4}$ | $0.04974$ | $1.052\cdot 10^{-4}$ | $0.05000$ | $2.121\cdot 10^{-4}$ | $0.1008$ | $11.69$
$5$ | $1\,609$ | $8.485\cdot 10^{-5}$ | $0.04960$ | $8.790\cdot 10^{-5}$ | $0.05139$ | $1.774\cdot 10^{-4}$ | $0.1037$ | $11.08$
$6$ | $9\,747$ | $7.056\cdot 10^{-5}$ | $0.05019$ | $7.371\cdot 10^{-5}$ | $0.05243$ | $1.488\cdot 10^{-4}$ | $0.1058$ | $10.65$
$7$ | $184\,607$ | $5.852\cdot 10^{-5}$ | $0.05397$ | $5.802\cdot 10^{-5}$ | $0.05351$ | $1.168\cdot 10^{-4}$ | $0.1077$ | $10.23$
$8$ | $1\,257\,728$ | $5.118\cdot 10^{-5}$ | $0.05411$ | $5.105\cdot 10^{-5}$ | $0.05397$ | $1.028\cdot 10^{-4}$ | $0.1087$ | $10.04$
$9$ | $2.290\cdot 10^{7}$ | $4.450\cdot 10^{-5}$ | $0.05620$ | $4.321\cdot 10^{-5}$ | $0.05457$ | $8.690\cdot 10^{-5}$ | $0.1097$ | $9.831$
$10$ | $1.560\cdot 10^{8}$ | $4.008\cdot 10^{-5}$ | $0.05609$ | $3.916\cdot 10^{-5}$ | $0.05480$ | $7.878\cdot 10^{-5}$ | $0.1102$ | $9.728$
$11$ | $3.910\cdot 10^{9}$ | $3.602\cdot 10^{-5}$ | $0.05792$ | $3.436\cdot 10^{-5}$ | $0.05526$ | $6.906\cdot 10^{-5}$ | $0.1110$ | $9.579$
$12$ | $2.740\cdot 10^{10}$ | $3.323\cdot 10^{-5}$ | $0.05774$ | $3.187\cdot 10^{-5}$ | $0.05538$ | $6.406\cdot 10^{-5}$ | $0.1113$ | $9.517$
$13$ | $7.560\cdot 10^{11}$ | $3.037\cdot 10^{-5}$ | $0.05914$ | $2.862\cdot 10^{-5}$ | $0.05574$ | $5.749\cdot 10^{-5}$ | $0.1120$ | $9.410$
$14$ | $5.430\cdot 10^{12}$ | $2.755\cdot 10^{-5}$ | $0.05899$ | $2.607\cdot 10^{-5}$ | $0.05582$ | $5.236\cdot 10^{-5}$ | $0.1121$ | $9.370$
$15$ | $1.610\cdot 10^{14}$ | $2.527\cdot 10^{-5}$ | $0.06010$ | $2.359\cdot 10^{-5}$ | $0.05610$ | $4.736\cdot 10^{-5}$ | $0.1126$ | $9.288$
$16$ | $1.170\cdot 10^{15}$ | $2.424\cdot 10^{-5}$ | $0.05987$ | $2.273\cdot 10^{-5}$ | $0.05614$ | $4.565\cdot 10^{-5}$ | $0.1127$ | $9.261$
$17$ | $3.700\cdot 10^{16}$ | $2.273\cdot 10^{-5}$ | $0.06080$ | $2.108\cdot 10^{-5}$ | $0.05638$ | $4.231\cdot 10^{-5}$ | $0.1132$ | $9.196$
$18$ | $2.730\cdot 10^{17}$ | $2.172\cdot 10^{-5}$ | $0.06062$ | $2.021\cdot 10^{-5}$ | $0.05639$ | $4.057\cdot 10^{-5}$ | $0.1132$ | $9.177$
$19$ | $9.030\cdot 10^{18}$ | $2.010\cdot 10^{-5}$ | $0.06140$ | $1.852\cdot 10^{-5}$ | $0.05660$ | $3.718\cdot 10^{-5}$ | $0.1136$ | $9.123$
$20$ | $6.740\cdot 10^{19}$ | $1.908\cdot 10^{-5}$ | $0.06122$ | $1.765\cdot 10^{-5}$ | $0.05661$ | $3.542\cdot 10^{-5}$ | $0.1136$ | $9.109$
$21+$ | $10^{n_{L}}$ | $1.819\cdot 10^{-5}$ | $0.06141$ | $1.679\cdot 10^{-5}$ | $0.05669$ | $3.370\cdot 10^{-5}$ | $0.1138$ | $9.082$

##  3\. The least prime in the Chebotarev density theorem - Proof of Theorem 1

###  3.1. Choosing a weight to detect the least prime

Let $x\geq 1$. Let $\theta>1$ be a parameter to be chosen later. We consider
the kernel

$k(s)=k_{\theta}(s)=\Big{(}\frac{x^{\theta(s-1)}-x^{s-1}}{s-1}\Big{)}^{2}$
(3.1)

and recall its inverse Mellin transform is

$\hat{k}(u)=\frac{1}{2\pi
i}\int_{2-i\infty}^{2+i\infty}k(s)u^{-s}ds=\left.\begin{cases}u^{-1}\log(x^{2\theta}/{u})&\text{if
}x^{\theta+1}\leq u\leq x^{2\theta};\\\ u^{-1}\log({u}/{x^{2}})&\text{if
}x^{2}\leq u\leq x^{\theta+1};\\\ 0&\text{otherwise.}\end{cases}\right.$ (3.2)

In [AhKw19-1, LaMoOd79], $\theta$ was chosen to be 2. Note that

$\displaystyle\hat{k}(u)\leq\frac{(\theta-1)(\log x)}{u},$ (3.3)
$\displaystyle k(1)=k_{\theta}(1)=((\theta-1)\log x)^{2},$ (3.4)
$\displaystyle
k(\sigma)=\Big{(}\frac{x^{\theta(\sigma-1)}-x^{\sigma-1}}{\sigma-1}\Big{)}^{2}=\Big{(}\frac{1-x^{(\theta-1)(\sigma-1)}}{\sigma-1}\Big{)}^{2}x^{2(\sigma-1)}\leq\frac{1}{(\sigma-1)^{2}}x^{2(\sigma-1)}\
\text{if}\ \sigma<1,$ (3.5)
$\displaystyle|k(\sigma+it)|\leq\frac{\Big{(}x^{\theta(\sigma-1)}+x^{\sigma-1}\Big{)}^{2}}{|s-1|^{2}}\leq
k(\sigma)\Big{(}\frac{1+x^{(\theta-1)(\sigma-1)}}{1-x^{(\theta-1)(\sigma-1)}}\Big{)}^{2}\frac{(\sigma-1)^{2}}{(\sigma-1)^{2}+t^{2}},$
(3.6) $\displaystyle k(1)-k(\beta_{1})=(\log
x)^{2}\phi_{\theta}((1-\beta_{1})\log x),\ \text{where}\
\phi_{\theta}(v)=(\theta-1)^{2}-\Big{(}\frac{e^{-v}-e^{-\theta
v}}{v}\Big{)}^{2}.$ (3.7)

We shall require the following lemma regarding the properties of
$\phi_{\theta}$.

######  Lemma 9.

Let $\theta>1$. Then

  1. (i)

$\phi_{\theta}(v)$ is increasing for $v>0$;

  2. (ii)

for any $v>0$, we have $\phi_{\theta}(v)\geq(\theta-1)^{2}(1-e^{-2v})$.
In particular, if $b>0$, then for any $0\leq v\leq b$, we have
$\phi_{\theta}(v)\geq 2(\theta-1)^{2}e^{-2b}v.$

###### Proof of Lemma 9.

We first note that

$\displaystyle\phi^{\prime}_{\theta}(v)$
$\displaystyle=-2\Big{(}\frac{e^{-v}-e^{-\theta
v}}{v}\Big{)}((-v^{-2})(e^{-v}-e^{-\theta v})+(v^{-1})(-e^{-v}+\theta
e^{-\theta v}))$ $\displaystyle=2\Big{(}\frac{e^{-v}-e^{-\theta
v}}{v^{2}}\Big{)}\Big{(}e^{-v}\Big{(}1+\frac{1}{v}\Big{)}-e^{-\theta
v}\Big{(}\theta+\frac{1}{v}\Big{)}\Big{)}.$

We claim $\phi^{\prime}_{\theta}(v)\geq 0$ for $v\geq 0$ and so
$\phi_{\theta}$ is increasing. More precisely, letting $f(\theta)=e^{-\theta
v}(\theta+\frac{1}{v})$, we claim $f(1)\geq f(\theta)$ whenever $\theta>1$.
This follows from $f^{\prime}(\theta)=-ve^{-\theta
v}(\theta+\frac{1}{v})+e^{-\theta v}=-ve^{-\theta v}\leq 0.$
Secondly, by the mean value theorem, for $\theta>1$, we have

$\Big{|}\frac{e^{-v}-e^{-\theta
v}}{1-\theta}\Big{|}=\max_{t\in[1,\theta]}|ve^{-tv}|\leq ve^{-v},$

and thus $\big{(}\frac{e^{-v}-e^{-\theta
v}}{v}\big{)}^{2}\leq(\theta-1)^{2}e^{-2v}.$ Hence, we obtain

$\phi_{\theta}(v)=(\theta-1)^{2}-\Big{(}\frac{e^{-v}-e^{-\theta
v}}{v}\Big{)}^{2}\geq(\theta-1)^{2}(1-e^{-2v}).$

Lastly, for $c>0$, letting $g_{c}(v)=(1-e^{-cv})-cve^{-cb}$, we note that
$g(v)\geq 0$ for all $v\in[0,b]$. Indeed, as $g_{c}(0)=0$ and
$g_{c}^{\prime}(v)=ce^{-cv}-ce^{-cb}\geq 0$ for all $v\leq b$, we know that
$g_{c}(v)$ is non-negative and increasing. Now, the last part of the lemma
follows from using $g_{c}(v)$ with $c=2$. ∎

###  3.2. Explicit inequalities

Throughout our discussion (from Section 3.2 to Section 3.4), we let $L/K$ be a
Galois extension of number fields with Galois group $G$, and we let $C$ be a
conjugacy class in $G$. We let $\mathcal{P}(C)$ denote the set of the
unramified primes $\mathfrak{p}\subset\mathcal{O}_{K}$ of degree one such that
$\sigma_{\mathfrak{p}}=C$. In addition, we choose $d_{0}$ according to Table 2
and assume $\log d_{L}\geq{\mathscr{L}}_{0}=\log d_{0}$. We let $x\geq 101$
and let $k$ be a weight to be chosen later. We recall that $\beta_{1}$ denotes
the possible exceptional zero, appearing in
$(1-\frac{1}{R_{0}{\mathscr{L}}},1)$, of $\zeta_{L}(s)$. By (2.3), we know
that there is at most one zero of $\zeta_{L}(s)$ in
$(1-\frac{1}{R_{1}{\mathscr{L}}},1-\frac{1}{R_{0}{\mathscr{L}}}]$; we shall
denote such a zero by $\beta^{\prime}$ if it exists.

Following [AhKw19-1] and [LaMoOd79], our goal is to show that there exists
$c_{4}>0$ such that for $x=d_{L}^{c_{4}}=e^{c_{4}{\mathscr{L}}}$, we have

$\mathcal{S}_{C}=\sum_{\mathfrak{p}\in\mathcal{P}(C)}(\log
N\mathfrak{p})\hat{k}(N\mathfrak{p})>0,$ (3.8)

which implies that there is a prime $\mathfrak{p}\in\mathcal{P}(C)$ such that

$N\mathfrak{p}\leq x^{2\theta}=d_{L}^{2\theta c_{4}}.$

Thus we want to estimate

$B=2\theta c_{4}.$ (3.9)

######  Proposition 10.

Let $k(s)$ be the kernel given as in (3.1), with $x=d_{L}^{c_{4}}\geq 101$,
and $\mathcal{S}_{C}$ be the sum given as in (3.8). Then we have

$\displaystyle\frac{|G|}{|C|}\mathcal{S}_{C}\geq$
$\displaystyle(1-\delta(\beta_{1}))(\theta-1)^{2}(\log
x)^{2}+\delta(\beta_{1})\phi_{\theta}((1-\beta_{1})\log x)(\log
x)^{2}-(1-\delta(\beta_{1}))\delta(\beta^{\prime})|k(\beta^{\prime})|$
$\displaystyle-\sum_{\rho\in\mathcal{S}\backslash\\{\beta_{1},\beta^{\prime},1-\beta_{1}\\}}|k(\rho)|-\alpha_{3}(\theta-1)\frac{|G|}{|C|}{\mathscr{L}}\frac{\log
x}{x}-\delta(\beta_{1})\frac{4}{(1-\frac{1}{2{\mathscr{L}}})^{2}}x^{-2(1-\frac{1}{2{\mathscr{L}}})}-c_{5}{\mathscr{L}}x^{-2},$

where $\phi_{\theta},\alpha_{3}$, and $c_{5}$ are defined in (3.7), (3.15),
and (3.20), respectively. Here $\delta(\beta_{1})=1$ only if the exceptional
zero $\beta_{1}$ exists for $\zeta_{L}(s)$, and $\delta(\beta_{1})=0$
otherwise. $\delta(\beta^{\prime})$ is defined the same way for $\beta_{1}$.

###### Proof.

Let $g_{C}\in G$ be a representative of $C$, and let

$\Psi_{C}(s)=-\frac{|C|}{|G|}\sum_{\chi}\overline{\chi}(g_{C})\frac{L^{\prime}}{L}(s,\chi,L/K),$
(3.10)

where the sum is over the irreducible characters $\chi$ of
$G=\operatorname{Gal}(L/K)$, and $L(s,\chi,L/K)$ is the Artin $L$-function
attached to $\chi$. It follows from the orthogonality property of irreducible
characters of $G$ that

$I_{C}=\frac{1}{2\pi
i}\int_{2-i\infty}^{2+i\infty}\Psi_{C}(s)k(s)ds=\sum_{\mathfrak{p}}\sum_{m=1}^{\infty}J(\mathfrak{p}^{m})(\log
N\mathfrak{p})\hat{k}(N\mathfrak{p}^{m}),$ (3.11)

where for ramified primes $\mathfrak{p}$, $|J(\mathfrak{p}^{m})|\leq 1$, for
unramified primes $\mathfrak{p}$, $J(\mathfrak{p}^{m})=1$ if
$\sigma_{\mathfrak{p}}^{m}=C$, and $J(\mathfrak{p}^{m})=0$ otherwise. By
(3.3), as argued in [AhKw19-1, Lemmata 3.1(1) and 3.3(1)] (see also [LaMoOd79,
Lemmata 3.1-3.3]), for $x\geq 2$, one has

$\sum_{\text{$\mathfrak{p}$ ramified}}\sum_{m\geq 1}J(\mathfrak{p}^{m})(\log
N\mathfrak{p})\hat{k}(N\mathfrak{p}^{m})\leq 2(\theta-1)\frac{\log
x}{x^{2}}\log d_{L}.$ (3.12)

Also, for $x\geq 101$, one has

$\sum_{\mathfrak{p},m}J(\mathfrak{p}^{m})(\log
N\mathfrak{p})\hat{k}(N\mathfrak{p}^{m})\leq
16.08\alpha_{0}(\theta-1)\frac{\log x}{x}n_{K},\ \text{with}\
\alpha_{0}=1.25506,$ (3.13)

where the sum is over $\mathfrak{p},m$ such that $N\mathfrak{p}^{m}\neq p$ for
any (rational) prime $p$. Hence, by Minkowski’s bound, for $x\geq 101$, we
have as in [AhKw19-1, Proposition 3.5]

$|I_{C}-\mathcal{S}_{C}|\leq\frac{2(\theta-1)\log x}{x^{2}}\log
d_{L}+16.08\alpha_{0}(\theta-1)n_{K}\frac{\log
x}{x}\leq\alpha_{3}(\theta-1)\frac{\log x}{x}\log d_{L},$ (3.14)

where

$\alpha_{3}=\frac{2}{101}+\frac{32.16\alpha_{0}}{\log 3}=36.7595\ldots.$
(3.15)

The second step is to relate $\Psi_{C}(s)$ to the zeros of the Dedekind zeta
function $\zeta_{L}(s)$. To do so, we recall that by Deuring’s reduction
[Deu35], denoting the fixed field of $g_{C}$ by $E$, one has

$\Psi_{C}(s)=-\frac{|C|}{|G|}\sum_{\psi}\overline{\psi}(g_{C})\frac{L^{\prime}}{L}(s,\psi,L/E),$

where the sum is over the irreducible characters $\psi$ of
$\operatorname{Gal}(L/E)$. As $L/E$ is abelian, it follows from Artin
reciprocity that each Artin $L$-function $L(s,\psi,L/E)$ corresponds to a
Hecke $L$-function. This allows us to use the classical method of contour
integration to find a lower bound of $I_{C}$ (in terms of $k(s)$ and the zeros
of $\zeta_{L}(s)$). Indeed, one has

$I_{C}=\frac{|C|}{|G|}\sum_{\psi}\overline{\psi}(g_{C})\Big{(}\frac{1}{2\pi
i}\int_{2-i\infty}^{2+i\infty}-\frac{L^{\prime}}{L}(s,\psi,L/E)k(s)ds\Big{)}.$

Proceeding exactly as in [AhKw19-1, Sec. 4] and [LaMoOd79, Sec. 3] (see also
[LaOd77]), an application of Cauchy’s integral formula gives

$\frac{|G|}{|C|}I_{C}\geq
k(1)-\sum_{\rho\in\mathcal{S}}|k(\rho)|-n_{L}k(0)-\sum_{\psi}|V(\psi)|,$
(3.16)

where

$V(\psi)=\frac{1}{2\pi
i}\int_{-\frac{1}{2}-i\infty}^{-\frac{1}{2}+i\infty}-\frac{L^{\prime}}{L}(s,\psi,L/E)k(s)ds.$

Denoting $A(\psi)$ the conductor of $L(s,\psi,L/E)$, we appeal to a bound for
the $L$-term of the form as given in [LaOd77, Lemma 6.2] and as given
explicitly in [Das21, Lemma 2.19]:

$\Big{|}\frac{L^{\prime}}{L}(-\frac{1}{2}+it,\psi,L/E)\Big{|}\leq\log
A(\psi)+n_{E}v(t),\ \text{with}\
v(t)=\log(\sqrt{0.25+t^{2}}+1)+4.452+\frac{83}{5},$ (3.17)

with $4.452+\frac{83}{5}=21.052$. (Note this is an improvement on [Win13,
Lemma 5.1] where instead $v(t)=\log(\sqrt{0.25+t^{2}}+2)+\frac{19683}{812}$.)

Using the bound (3.6) for $k$, we have for any $x\geq 101$,

$\Big{|}k\Big{(}-\frac{1}{2}+it\Big{)}\Big{|}\leq
k\Big{(}-\frac{1}{2}\Big{)}\Big{(}\frac{1+101^{-\frac{3}{2}(\theta-1)}}{1-101^{-\frac{3}{2}(\theta-1)}}\Big{)}^{2}\frac{9}{9+4t^{2}}.$

Hence, we deduce

$|V(\psi)|\leq\frac{1}{2\pi}k\Big{(}-\frac{1}{2}\Big{)}\Big{(}\frac{1+101^{-\frac{3}{2}(\theta-1)}}{1-101^{-\frac{3}{2}(\theta-1)}}\Big{)}^{2}\int_{-\infty}^{\infty}(\log
A(\psi)+n_{E}v(t))\frac{9}{9+4t^{2}}dt.$

Now, from (3.16) and the conductor-discriminant formula $\sum_{\psi}\log
A(\psi)=\log d_{L}$,

$\frac{|G|}{|C|}I_{C}\geq
k(1)-\sum_{\rho\in\mathcal{S}}|k(\rho)|-W_{0}k\Big{(}-\frac{1}{2}\Big{)}\log
d_{L}-n_{L}\Big{(}k(0)+W_{1}k\Big{(}-\frac{1}{2}\Big{)}\Big{)},$ (3.18)

where, as before, $\mathcal{S}$ denotes the set of all non-trivial zeros of
$\zeta_{L}(s)$,

$W_{0}=\frac{1}{\pi}\int_{0}^{\infty}W(t)dt,\enspace
W_{1}=\frac{1}{\pi}\int_{0}^{\infty}v(t)W(t)dt,\enspace
W(t)=\Big{(}\frac{1+101^{-\frac{3}{2}(\theta-1)}}{1-101^{-\frac{3}{2}(\theta-1)}}\Big{)}^{2}\frac{9}{9+4t^{2}},$
(3.19)

and $v(t)$ is defined in (3.17). Together with Minkowski’s bound and (3.5), we
deduce, for $x\geq 101$,

$W_{0}k\Big{(}-\frac{1}{2}\Big{)}\log
d_{L}+n_{L}\Big{(}k(0)+W_{1}k\Big{(}-\frac{1}{2}\Big{)}\Big{)}\leq c_{5}(\log
d_{L})x^{-2},$

where

$c_{5}=\frac{2}{\log 3}+\frac{4}{909}\Big{(}W_{0}+\frac{2}{\log
3}W_{1}\Big{)}.$ (3.20)

Hence, putting (3.14) and (3.18) together, if $\zeta_{L}(s)$ has no
exceptional zero, we have

$\frac{|G|}{|C|}\mathcal{S}_{C}\geq
k(1)-\sum_{\rho\in\mathcal{S}}|k(\rho)|-\frac{c_{5}{\mathscr{L}}}{x^{2}}-\alpha_{3}(\theta-1)\frac{|G|}{|C|}\frac{\log
x}{x}{\mathscr{L}},$ (3.21)

where $c_{5}$ and $\alpha_{3}$ are defined in (3.20) and (3.15). We then use
(3.4) to bound $k(1)$. Otherwise, if $\zeta_{L}(s)$ admits an exceptional zero
$\beta_{1}$, the term $k(1)-k(\beta_{1})$ appears. We use (3.7) to control the
term arising at $\beta_{1}$ and (3.5) at $1-\beta_{1}$. Thus

$k(1)-k(\beta_{1})=(\log x)^{2}\phi_{\theta}((1-\beta_{1})\log x),\
\text{and}\
|k(1-\beta_{1})|\leq\frac{4x^{-2\beta_{1}}}{\beta_{1}^{2}}\leq\frac{4x^{-2(1-\frac{1}{2{\mathscr{L}}})}}{(1-\frac{1}{2{\mathscr{L}}})^{2}}.$
(3.22)

Thus, in the case the exceptional zero $\beta_{1}$ appears, we have

$\displaystyle\begin{split}\frac{|G|}{|C|}\mathcal{S}_{C}\geq&(\log
x)^{2}\phi_{\theta}((1-\beta_{1})\log
x)+\frac{4}{(1-\frac{1}{2{\mathscr{L}}})^{2}}x^{-2(1-\frac{1}{2{\mathscr{L}}})}\\\
&-\sum_{\rho\in\mathcal{S}\backslash\\{\beta_{1},1-\beta_{1}\\}}|k(\rho)|-\frac{c_{5}{\mathscr{L}}}{x^{2}}-\alpha_{3}(\theta-1)\frac{|G|}{|C|}\frac{\log
x}{x}{\mathscr{L}}.\end{split}$ (3.23)

Finally, writing the sum in (3.23) as

$\sum_{\rho\in\mathcal{S}\backslash\\{\beta_{1},1-\beta_{1}\\}}|k(\rho)|=(1-\delta(\beta_{1}))\delta(\beta^{\prime})|k(\beta^{\prime})|+\sum_{\rho\in\mathcal{S}\backslash\\{\beta_{1},\beta^{\prime}1-\beta_{1}\\}}|k(\rho)|,$

we conclude the proof. ∎

We now focus on controlling the size of the above sums over the zeros. This is
done by means of zero-free regions, Deuring-Heilbronn repulsion, and zero-
density estimates for zeros of $\zeta_{L}(s)$. First, we have some explicit
estimate for $N_{L}(T)$ which counts the number of non-trivial zeros
$\rho=\beta+i\gamma$ of $\zeta_{L}(s)$ such that $|\gamma|\leq T$. We refer to
the recent improvement of [HaShWo]:

######  Lemma 11.

Let $T\geq 1$. Then

$\Big{|}N_{L}(T)-\frac{T}{\pi}\log\Big{(}d_{L}\Big{(}\frac{T}{2\pi
e}\Big{)}^{n_{L}}\Big{)}\Big{|}\leq 0.296(\log d_{L}+n_{L}\log
T)+3.971n_{L}+3.969.$ (3.24)

In addition, we require a sharper estimate for the number of non-trivial zeros
close to $1$. Denoting $Z(r;s)=\\{\rho\in\mathcal{S}\mid|\rho-s|\leq r\\}$ ,
we consider $n(r;s)=|Z(r;s)|$, the number of non-trivial zeros $\rho$ of
$\zeta_{L}(s)$ such that $|\rho-s|\leq r$.

######  Lemma 12.

Let $n_{0}\geq 2,\mathscr{Q}_{0}>0,0<r\leq 1$, and $\alpha>0$. If $n_{0}\leq
n_{L}\leq\mathscr{Q}_{0}{\mathscr{L}}$, then

$n(r;1)\leq\Big{(}\frac{1+\alpha}{\alpha}\Big{)}^{2}(1+\alpha
r\omega(\alpha){\mathscr{L}}),$ (3.25)

where

$\omega(\alpha)=\frac{1}{2}+\frac{\mathscr{Q}_{0}}{2}\max\Big{\\{}\frac{\Gamma^{\prime}}{\Gamma}\Big{(}\frac{2+\alpha}{2}\Big{)}-\log\pi+\frac{2}{n_{0}},0\Big{\\}}.$
(3.26)

###### Proof.

We let $0<r\leq 1$ and $\alpha>0$ and set $s_{0}=1+\alpha r$. Putting together
the classical explicit formula (see, e.g., [AhKw19-1, p. 1442])

$\sum_{\rho\in\mathcal{S}}{\mathfrak{Re}}\frac{1}{s_{0}-\rho}=\frac{1}{2}\log
d_{L}+{\mathfrak{Re}}\Big{(}\frac{1}{s}_{0}+\frac{1}{s_{0}-1}\Big{)}+{\mathfrak{Re}}\frac{\gamma^{\prime}_{L}}{\gamma_{L}}(s_{0})+{\mathfrak{Re}}\frac{\zeta^{\prime}_{L}}{\zeta_{L}}(s_{0}),$
(3.27)

with the facts that
${\mathfrak{Re}}\frac{\zeta^{\prime}_{L}}{\zeta_{L}}(s_{0})=\frac{\zeta^{\prime}_{L}}{\zeta_{L}}(1+\alpha
r)\leq 0,$ and

$\frac{\gamma^{\prime}_{L}}{\gamma_{L}}(s_{0})=\frac{(r_{1}+r_{2})}{2}\frac{\Gamma^{\prime}}{\Gamma}\Big{(}\frac{s_{0}}{2}\Big{)}+\frac{r_{2}}{2}\frac{\Gamma^{\prime}}{\Gamma}\Big{(}\frac{s_{0}+1}{2}\Big{)}-\frac{n_{L}}{2}\log\pi\leq\frac{n_{L}}{2}\Big{(}\frac{\Gamma^{\prime}}{\Gamma}\Big{(}\frac{2+\alpha}{2}\Big{)}-\log\pi\Big{)},$

yields

$\sum_{\rho\in\mathcal{S}}{\mathfrak{Re}}\frac{1}{s_{0}-\rho}\leq\frac{1}{\alpha
r}+\frac{1}{2}\log
d_{L}+\frac{n_{L}}{2}\Big{(}\frac{\Gamma^{\prime}}{\Gamma}\Big{(}\frac{2+\alpha}{2}\Big{)}-\log\pi\Big{)}+1.$

Note that

$\sum_{\rho\in\mathcal{S}}{\mathfrak{Re}}\frac{1}{s_{0}-\rho}\geq\sum_{\rho\in
Z((1+\alpha)r;s_{0})}{\mathfrak{Re}}\frac{1}{s_{0}-\rho}\geq\frac{\alpha
r}{((1+\alpha)r)^{2}}n((1+\alpha)r;s_{0})=\frac{\alpha}{(1+\alpha)^{2}r}n((1+\alpha)r;s_{0}).$

Observing that $Z(r;1)\subseteq Z((1+\alpha)r;1+\alpha
r)=Z((1+\alpha)r;s_{0})$, (3.25) follows from

$n(r;1)\leq
n((1+\alpha)r;s_{0})\leq\frac{(1+\alpha)^{2}r}{\alpha}\Big{(}\frac{1}{\alpha
r}+\frac{1}{2}\log
d_{L}+\frac{n_{L}}{2}\Big{(}\frac{\Gamma^{\prime}}{\Gamma}\Big{(}\frac{2+\alpha}{2}\Big{)}-\log\pi\Big{)}+1\Big{)}.\qed$

In addition, we require the following result on sums over the zeros close to
$1$.

######  Corollary 12.1.

Assume (1.5), $\alpha>0,r>0$ and that $\frac{1}{r{\mathscr{L}}}\leq 1$. If
$n_{0}\leq n_{L}\leq\mathscr{Q}_{0}{\mathscr{L}}$, then

$\sum_{\frac{1}{r{\mathscr{L}}}\leq|\rho-1|\leq
1}\frac{1}{|\rho-1|^{2}}\leq\Big{(}\Big{(}\frac{1+\alpha}{\alpha}\Big{)}^{2}\Big{(}r^{2}+2r\alpha\omega(\alpha)\Big{)}-n(\frac{1}{r{\mathscr{L}}};1)r^{2}\Big{)}{\mathscr{L}}^{2},$
(3.28)

where $\omega(\alpha)$, depending on $n_{0}$ and $\mathscr{Q}_{0}$, is defined
as in (3.26).

###### Proof.

We start with

$\sum_{\frac{1}{r{\mathscr{L}}}\leq|\rho-1|\leq
1}\frac{1}{|\rho-1|^{2}}=\int_{\frac{1}{r{\mathscr{L}}}}^{1}\frac{1}{u^{2}}dn(u;1)=n(1;1)-n\Big{(}\frac{1}{r{\mathscr{L}}};1\Big{)}(r{\mathscr{L}})^{2}+2\int_{\frac{1}{r{\mathscr{L}}}}^{1}\frac{n(u;1)}{u^{3}}du.$
(3.29)

It follows from Lemma 12 that

$n(1;1)\leq\Big{(}\frac{1+\alpha}{\alpha}\Big{)}^{2}(\alpha\omega(\alpha){\mathscr{L}}+1),\text{and}\int_{\frac{1}{r{\mathscr{L}}}}^{1}\frac{1+\alpha
u\omega(\alpha){\mathscr{L}}}{u^{3}}du=\Big{(}\frac{r^{2}}{2}+\alpha\omega(\alpha)r\Big{)}{\mathscr{L}}^{2}-\alpha\omega(\alpha){\mathscr{L}}-\frac{1}{2}.$
(3.30)

We conclude by putting together (3.29) and (3.30). ∎

We now control non-trivial zeros $\rho=\beta+i\gamma$ of $\zeta_{L}(s)$ that
are not equal to $\beta_{1}$, $\beta^{\prime}$, or $1-\beta_{1}$.

######  Proposition 13.

Let $k(s)$ be the kernel given as in (3.1), with $x=d_{L}^{c_{4}}$, and
$\mathcal{S}_{C}$ be the sum given as in (3.8). Let $R_{1}$ be defined as in
(2.3). If $n_{0}\leq n_{L}\leq\mathcal{Q}_{0}{\mathscr{L}}$, then

$\displaystyle\frac{|G|}{|C|}\mathcal{S}_{C}\geq$
$\displaystyle(1-\delta(\beta_{1}))(\theta-1)^{2}c_{4}^{2}{\mathscr{L}}^{2}+\delta(\beta_{1})\phi_{\theta}\Big{(}c_{4}(1-\beta_{1}){\mathscr{L}}\Big{)}c_{4}^{2}{\mathscr{L}}^{2}$
$\displaystyle-(1-\delta(\beta_{1}))\delta(\beta^{\prime})|k(\beta^{\prime})|-c_{6}{\mathscr{L}}-\sideset{}{{}^{\prime}}{\sum}_{\frac{1}{R_{1}{\mathscr{L}}}\leq|\rho-1|\leq
1}|k(\rho)|$
$\displaystyle-{\mathscr{Q}}_{0}\alpha_{3}(\theta-1)c_{4}{\mathscr{L}}^{3}e^{-c_{4}{\mathscr{L}}}-\delta(\beta_{1})\frac{4}{(1-\frac{1}{2{\mathscr{L}}})^{2}}e^{-c_{4}(2{\mathscr{L}}-1)}-c_{5}{\mathscr{L}}e^{-2c_{4}{\mathscr{L}}},$

where $\delta(\beta_{1})=1$ (resp., $\delta(\beta^{\prime})=1$) only if the
exceptional zero $\beta_{1}$ (resp., real zero $\beta^{\prime}$) exists for
$\zeta_{L}(s)$, $\delta(\beta_{1})=0$ (resp., $\delta(\beta^{\prime})=0$)
otherwise, $\phi_{\theta},\alpha_{3}$, $c_{5}$ and $c_{6}$ are defined in
(3.7), (3.15), (3.20), and (3.33), respectively, and, as later, the primed sum
is over non-trivial zeros $\rho\neq 1-\beta_{1}$ of $\zeta_{L}(s)$.

###### Proof.

We shall note that if $\rho=\beta+i\gamma$ is such that
$|\rho-1|<\frac{1}{R_{1}{\mathscr{L}}}$, then
$1-\beta<\frac{1}{R_{1}{\mathscr{L}}}$ and
$|\gamma|<\frac{1}{R_{1}{\mathscr{L}}}$. Hence, (2.3) forces that $\rho$ is
either $\beta_{1}$ or $\beta^{\prime}$ if
$|\rho-1|<\frac{1}{R_{1}{\mathscr{L}}}$. Thus, we can consider the splitting

$\sum_{\rho\in\mathcal{S}\backslash\\{\beta_{1},\beta^{\prime},1-\beta_{1}\\}}|k(\rho)|=\sideset{}{{}^{\prime}}{\sum}_{\frac{1}{R_{1}{\mathscr{L}}}\leq|\rho-1|\leq
1}|k(\rho)|+\sum_{|\rho-1|>1}|k(\rho)|.$ (3.31)

To bound $\sum_{|\rho-1|>1}|k(\rho)|$, we appeal to (3.6) to bound $k$:

$|k(\rho)|\leq\frac{(x^{\theta(\beta-1)}+x^{\beta-1})^{2}}{|\rho-1|^{2}}\leq\frac{4}{|\rho-1|^{2}}.$
(3.32)

Recalling that $n(r;1)$ denotes the number of non-trivial zeros $\rho$ of
$\zeta_{L}(s)$ such that $|\rho-1|\leq r$, we notice that $n(r;1)\leq
N_{L}(r)$. Using Lemma 12, we derive

$\sum_{|\rho-1|>1}|k(\rho)|\leq 4\int_{1}^{\infty}\frac{1}{r^{2}}dn(1;r)\leq
8\int_{1}^{\infty}\frac{n(1;r)}{r^{3}}dr\leq c_{6}{\mathscr{L}},$

where

$c_{6}=8\Big{(}\frac{1}{\pi}+0.148+{\mathscr{Q}}_{0}\max\Big{\\{}0,\int_{1}^{\infty}\frac{\frac{r}{\pi}\log(\frac{r}{2\pi
e})+0.296\log r+3.971+\frac{3.969}{n_{0}}}{r^{3}}dr\Big{\\}}\Big{)}.$ (3.33)

We conclude by putting together Proposition 10 with the above and the facts
that $\frac{|G|}{|C|}\leq n_{L}$ and $x=e^{c_{4}{\mathscr{L}}}$. ∎

Now, it remains to bound
$\sum^{\prime}_{\frac{1}{R_{1}{\mathscr{L}}}\leq|\rho-1|\leq 1}|k(\rho)|$. We
shall proceed with the argument by considering cases depending on the
existence of the exceptional zero $\beta_{1}$ and the distance of the
exceptional zero $\beta_{1}$, if it exists, to the $1$-line.

###  3.3. Non-exceptional case

######  Proposition 14.

Let $\mathcal{S}_{C}$ be the sum given as in (3.8). Assume there is no
exceptional zero $\beta_{1}$ in $(1-\frac{1}{R_{0}{\mathscr{L}}},1)$. If
$n_{0}\leq n_{L}\leq\mathcal{Q}_{0}{\mathscr{L}}$ and
${\mathscr{L}}\geq{\mathscr{L}}_{0}$, then

${\mathscr{L}}^{-2}\frac{|G|}{|C|}\mathcal{S}_{C}\geq(\theta-1)^{2}c_{4}^{2}-c_{7}(c_{4})e^{\frac{-2c_{4}}{29.57(1+\Delta_{0}(1))}}-\mathcal{E}_{0}(c_{4}),$
(3.34)

where

$\mathcal{E}_{0}(t)=\frac{c_{6}}{{\mathscr{L}}_{0}}+\frac{R_{0}^{2}}{{\mathscr{L}}_{0}^{2}}e^{-\frac{2t}{R_{0}}{\mathscr{L}}_{0}}+\mathscr{Q}_{0}\alpha_{3}(\theta-1)c_{4}{\mathscr{L}}_{0}e^{-c_{4}{\mathscr{L}}_{0}}+\frac{c_{5}}{{\mathscr{L}}_{0}}e^{-2c_{4}{\mathscr{L}}_{0}},$
(3.35)

and $c_{5}$, $c_{6}$, and $c_{7}$ are defined as in (3.20), (3.33), and
(3.37), respectively.

###### Proof.

We begin by recalling that $\beta^{\prime}\leq 1-\frac{1}{R_{0}{\mathscr{L}}}$
(if exists) and that for any non-trivial zero
$\rho=\beta+i\gamma\neq\beta^{\prime}$, with $|\gamma|\leq 1$, the zero-free
region [AhKw19-1, Proposition 6.1] gives

$1-\beta>(29.57\log(d_{L}\tau^{n_{L}}))^{-1}\geq(29.57{\mathscr{L}}(1+\delta_{L}(3)))^{-1}\geq(29.57{\mathscr{L}}(1+\Delta_{0}(1))^{-1},$

where $\delta_{L}$ and $\Delta_{0}$ are defined in (1.6) and (1.7),
respectively, and the last inequality is due to (1.9).

Therefore, by (3.5), we have

$|k(\beta^{\prime})|\leq\frac{1}{(\beta^{\prime}-1)^{2}}x^{2(\beta^{\prime}-1)}\leq
R_{0}^{2}x^{-\frac{2}{R_{0}}},$

and by using the first bound of (3.6), we have for the other zeros that,

$|k(\rho)|\leq\frac{x^{2(\beta-1)}(1+x^{(\theta-1)(\beta-1)})^{2}}{|\rho-1|^{2}}\leq\frac{x^{-2(29.57{\mathscr{L}}(1+\Delta_{0}(1)))^{-1}}(1+x^{(1-\theta)(29.57{\mathscr{L}}(1+\Delta_{0}(1)))^{-1}})^{2}}{|\rho-1|^{2}}.$

By Corollary 12.1, $n(\frac{1}{2{\mathscr{L}}};1)=0$ and putting
$x=e^{c_{4}{\mathscr{L}}}$, we then derive

$(1-\delta(\beta_{1}))\delta(\beta^{\prime})|k(\beta^{\prime})|+\sideset{}{{}^{\prime}}{\sum}_{\frac{1}{2{\mathscr{L}}}\leq|\rho-1|\leq
1}|k(\rho)|\leq
R_{0}^{2}e^{-\frac{2c_{4}}{R_{0}}{\mathscr{L}}}+c_{7}(c_{4}){\mathscr{L}}^{2}e^{\frac{-2c_{4}}{29.57(1+\Delta_{0}(1))}},$
(3.36)

where

$c_{7}(t)=4(1+\alpha\omega(\alpha))\Big{(}\frac{1+\alpha}{\alpha}\Big{)}^{2}\Big{(}1+e^{\frac{(1-\theta)t}{29.57(1+\Delta_{0}(1))}}\Big{)}^{2},$
(3.37)

and $\omega(\alpha)$ is defined as in (3.26). For
$n_{L}\leq\mathscr{Q}_{0}{\mathscr{L}}$ , we apply Proposition 13 with
$R_{1}=2$ and (3.36) to obtain

$\displaystyle\frac{|G|}{|C|}\mathcal{S}_{C}\geq$
$\displaystyle(\theta-1)^{2}c_{4}^{2}{\mathscr{L}}^{2}-c_{6}{\mathscr{L}}-R_{0}^{2}e^{-\frac{2c_{4}}{R_{0}}{\mathscr{L}}}-c_{7}{\mathscr{L}}^{2}e^{\frac{-2c_{4}}{29.57(1+\Delta_{0}(1))}}$
$\displaystyle-\mathscr{Q}_{0}\alpha_{3}(\theta-1)c_{4}{\mathscr{L}}^{3}e^{-c_{4}{\mathscr{L}}}-c_{5}{\mathscr{L}}e^{-2c_{4}{\mathscr{L}}}.$

We conclude with the fact that ${\mathscr{L}}\geq{\mathscr{L}}_{0}$. ∎

###  3.4. Exceptional case

Assume that the exceptional zero $\beta_{1}$ of $\zeta_{L}(s)$ presents such
that $\beta_{1}\geq 1-\frac{1}{R_{0}{\mathscr{L}}}$. Following Proposition 6,
we take $R_{0}=20$ and $R_{1}=1.24$. We recall $R=29.57$. We let
$\varepsilon_{1}>0$, $\sigma_{1}\geq 1$, and $\eta\in(0,1]$ be parameters (to
be chosen later) to compute $c_{1}^{\prime}(\varepsilon_{1},\sigma_{1},\eta)$
and $c_{2}^{\prime}(\varepsilon_{1},\sigma_{1},\eta)$ defined in (2.28). These
parameters will be chosen to make $c_{4}$ and thus $B$ as small as possible.

######  Proposition 15.

Let $\mathcal{S}_{C}$ be the sum given as in (3.8). Assume that the
exceptional zero $\beta_{1}$ of $\zeta_{L}(s)$ exists, that is $\beta_{1}\geq
1-\frac{1}{R_{0}{\mathscr{L}}}$. Let $\eta\in(0,1]$ satisfying

$\eta\geq
c_{2}^{\prime}(\varepsilon_{1},\sigma_{1},\eta)\frac{\log\Big{(}\frac{c_{1}^{\prime}(\varepsilon_{1},\sigma_{1},\eta)}{(1-\beta_{1}){\mathscr{L}}}\Big{)}}{{\mathscr{L}}}$
(3.38)

(with $c_{1}^{\prime}(\varepsilon_{1},\sigma_{1},\eta)$ and
$c_{2}^{\prime}(\varepsilon_{1},\sigma_{1},\eta)$ defined in (2.28)). If
$n_{0}\leq n_{L}\leq\mathscr{Q}_{0}{\mathscr{L}}$ and
${\mathscr{L}}\geq{\mathscr{L}}_{0}$, then

$\begin{split}{\mathscr{L}}^{-2}\frac{|G|}{|C|}\mathcal{S}_{C}\geq&\phi_{\theta}((1-\beta_{1})c_{4}{\mathscr{L}})c_{4}^{2}-c_{11}((1-\beta_{1}){\mathscr{L}})^{2c_{4}c_{8}}-\frac{c_{6}}{{\mathscr{L}}}\\\
&-{\mathscr{Q}}_{0}\alpha_{3}(\theta-1)c_{4}{\mathscr{L}}e^{-c_{4}{\mathscr{L}}}-\frac{4}{({\mathscr{L}}-\frac{1}{2})^{2}}e^{-c_{4}(2{\mathscr{L}}-1)}-\frac{c_{5}}{{\mathscr{L}}}e^{-2c_{4}{\mathscr{L}}},\end{split}$
(3.39)

where $\alpha_{3}$, $c_{5}$, $c_{6}$, $c_{8}$, and $c_{11}$ are defined in
(3.15), (3.20), (3.33), (3.42), and (3.45), respectively.

###### Proof.

We start by estimating the sum over low-lying zeros
$\sum^{\prime}_{\frac{1}{R_{1}{\mathscr{L}}}\leq|\rho-1|\leq 1}|k(\rho)|$. Let
$\rho=\beta+it$ denote a non-exceptional zero of $\zeta_{L}(s)$ such that
$\frac{1}{R_{1}{\mathscr{L}}}\leq|\rho-1|\leq 1$. We shall consider the
following two situations:

  1. (a)

We first suppose that $(1-\beta_{1}){\mathscr{L}}\leq
c_{1}^{\prime}(\varepsilon_{1},\sigma_{1},\eta)^{a}$. Applying Theorem 2, we
have either $\beta\leq 1-\eta$ or

$1-\beta\geq
c_{2}^{\prime}(\varepsilon_{1},\sigma_{1},\eta)\frac{\log\Big{(}\frac{c_{1}^{\prime}(\varepsilon_{1},\sigma_{1},\eta)}{(1-\beta_{1}){\mathscr{L}}}\Big{)}}{{\mathscr{L}}}\geq
c_{9}\frac{\log\Big{(}\frac{1}{(1-\beta_{1}){\mathscr{L}}}\Big{)}}{{\mathscr{L}}},$

where

$c_{9}=c_{2}^{\prime}(\varepsilon_{1},\sigma_{1},\eta)\Big{(}1-\frac{1}{a}\Big{)}.$

  2. (b)

Secondly, we consider the situation that
$(1-\beta_{1}){\mathscr{L}}>c_{1}^{\prime}(\varepsilon_{1},\sigma_{1},\eta)^{a}$.
By [AhKw19-1, Proposition 6.1] and (1.9), we know that

$1-\beta>(29.57\log(d_{L}\tau^{n_{L}}))^{-1}\geq(29.57{\mathscr{L}}(1+\Delta_{0}(1)))^{-1}$

as $\tau=|t|+2$ and $|t|\leq 1$. Thus, by choosing

$c_{10}=\frac{1}{a\cdot
29.57(1+\Delta_{0}(1))\log(1/c_{1}^{\prime}(\varepsilon_{1},\sigma_{1},\eta))},$

we have

$1-\beta\geq
c_{10}\frac{1}{{\mathscr{L}}}\log\Big{(}\frac{1}{c_{1}^{\prime}(\varepsilon_{1},\sigma_{1},\eta)^{a}}\Big{)}\geq
c_{10}\frac{\log\Big{(}\frac{1}{(1-\beta_{1}){\mathscr{L}}}\Big{)}}{{\mathscr{L}}}.$

To summarize, for any non-exceptional zero $\rho=\beta+it$ such that
$\frac{1}{R_{1}{\mathscr{L}}}\leq|\rho-1|\leq 1$, assuming (3.38), we have

$1-\beta\geq
B:=c_{8}\frac{\log\Big{(}\frac{1}{(1-\beta_{1}){\mathscr{L}}}\Big{)}}{{\mathscr{L}}},\
\text{with}\ c_{8}=\min\\{c_{9},c_{10}\\}.$ (3.40)

We chose $a$ such that $c_{9}=c_{10}$, i.e.

$a=1+\frac{1}{29.57(1+\Delta_{0}(1))c_{2}^{\prime}(\varepsilon_{1},\sigma_{1},\eta)\log(1/c_{1}^{\prime}(\varepsilon_{1},\sigma_{1},\eta))}.$
(3.41)

Thus, $c_{8}$ is given by

$c_{8}=\frac{1}{1/c_{2}^{\prime}(\varepsilon_{1},\sigma_{1},\eta)+29.57(1+\Delta_{0}(1))\log(1/c_{1}^{\prime}(\varepsilon_{1},\sigma_{1},\eta))}.$
(3.42)

Applying the bounds (3.6) on $k$ with $1-\beta\geq B$ gives

$|k(\rho)|\leq\frac{x^{2(\beta-1)}(1+x^{(\theta-1)(\beta-1)})^{2}}{|\rho-1|^{2}}\leq\frac{x^{-2B}(1+x^{-(\theta-1)B})^{2}}{|\rho-1|^{2}}.$
(3.43)

Recalling that we set $x=d_{L}^{c_{4}}$, and by the definition of the
exceptional zero with respect to the zero-free region (2.2), that $B\geq
c_{8}\frac{\log R_{0}}{{\mathscr{L}}}$, we have

$x^{-2B}=((1-\beta_{1}){\mathscr{L}})^{2c_{4}c_{8}},\ \text{and}\
(1+x^{-(\theta-1)B})^{2}\leq\Big{(}1+e^{-(\theta-1)c_{4}c_{8}\log
R_{0}}\Big{)}^{2}.$

We apply Corollary 12.1 with $n(\frac{1}{R_{1}{\mathscr{L}}};1)=1$:

$\sideset{}{{}^{\prime}}{\sum}_{\frac{1}{R_{1}{\mathscr{L}}}\leq|\rho-1|\leq
1}\frac{1}{|\rho-1|^{2}}\leq\Big{(}\Big{(}\frac{1+\alpha}{\alpha}\Big{)}^{2}(R_{1}^{2}+2\alpha\omega(\alpha))R_{1}-R_{1}^{2}\Big{)}{\mathscr{L}}^{2},$

which together with (3.43), gives

$\sideset{}{{}^{\prime}}{\sum}_{\frac{1}{R_{1}{\mathscr{L}}}\leq|\rho-1|\leq
1}|k(\rho)|\leq
c_{11}{\mathscr{L}}^{2}((1-\beta_{1}){\mathscr{L}})^{2c_{4}c_{8}},$ (3.44)

with

$c_{11}=\Big{(}1+e^{-(\theta-1)c_{4}c_{8}\log
R_{0}}\Big{)}^{2}R_{1}\Big{(}\Big{(}\frac{1+\alpha}{\alpha}\Big{)}^{2}(R_{1}+2\alpha\omega(\alpha))-R_{1}\Big{)},$
(3.45)

where $\omega(\alpha)$, depending on $\mathscr{Q}_{0}$ and $n_{0}$, is defined
in (3.26). Note that $c_{9},c_{10}$, and $c_{11}$ all depend on
$\varepsilon_{1},\sigma_{1}$, and $\eta$. We conclude by using Proposition 13
and (3.44). ∎

For the rest of the article, $n_{0}\leq n_{L}\leq\mathcal{Q}_{0}{\mathscr{L}}$
and ${\mathscr{L}}\geq{\mathscr{L}}_{0}$, with ${\mathscr{L}}_{0}$ and
$\mathcal{Q}_{0}$ as defined in (1.4) and (1.8), respectively. For each
$(n_{0},d_{0})$ listed in Table 2, we choose $\varepsilon_{1}>0,\sigma_{1}\geq
1$, and $\eta\in(0,1]$ which give
$c_{1}^{\prime}(\varepsilon_{1},\sigma_{1},\eta)$ and
$c_{2}^{\prime}(\varepsilon_{1},\sigma_{1},\eta)$ as defined in (2.28). We
then obtain $c_{8}$ as defined in (3.42), and define

$c_{4}=\frac{1}{2c_{8}}+0.001.$ (3.46)

To prove Theorem 1, we shall split our consideration depending on the distance
of $\beta_{1}$ to 1-line. We shall introduce further parameters:

$\varepsilon_{2}>0,\ \sigma_{2}\geq 1,\ \kappa\geq 1,\ 0<\lambda\leq 1,\
0<\mu\leq 1,\ \text{and}\ 1<\nu\leq 2.$

For the rest of the article, we denote

$C_{1}=c_{1}^{\prime}(\varepsilon_{2},\sigma_{2},0.5)\ \text{and}\
C_{2}=c_{2}^{\prime}(\varepsilon_{2},\sigma_{2},0.5).$ (3.47)

We assume

$\frac{(\kappa
C_{1})^{2}}{{\mathscr{L}}}<\frac{\mu}{c_{4}{\mathscr{L}}^{\nu-1}}<\frac{\lambda}{c_{4}}<\frac{1}{R_{0}},$

where $C_{1}$ is defined in (2.28). We explore the cases:

  * •

“medium” when
$\frac{\lambda}{c_{4}}\leq(1-\beta_{1}){\mathscr{L}}<\frac{1}{R_{0}}$,

  * •

“small” when
$\frac{\mu}{c_{4}{\mathscr{L}}^{\nu-1}}\leq(1-\beta_{1}){\mathscr{L}}\leq\frac{\lambda}{c_{4}}$,

  * •

“very small” when $\frac{(\kappa
C_{1})^{2}}{{\mathscr{L}}}\leq(1-\beta_{1}){\mathscr{L}}\leq\frac{\mu}{c_{4}{\mathscr{L}}^{\nu-1}}$,
and

  * •

“extremely small” when $(1-\beta_{1}){\mathscr{L}}<\frac{(\kappa
C_{1})^{2}}{{\mathscr{L}}}$.

For the “extremely small” case, we will use a different weight $k$ and thus
leave this for the end.
In each of the first three cases, we use the weight $k_{\theta}$ as defined in
(3.1), and obtain $B=2\theta c_{4}$ as given by (3.9). The choice of
parameters will ensure $B$ to be as small as possible while keeping the
expression in the right of (3.39) positive. We note that if
$(1-\beta_{1}){\mathscr{L}}\geq\frac{(\kappa C_{1})^{2}}{{\mathscr{L}}}$, then
the condition (3.38), which is needed to apply Proposition 15, is satisfied
when

$\frac{c_{2}^{\prime}(\varepsilon_{1},\sigma_{1},\eta)}{{\mathscr{L}}_{0}}\Big{(}\max\Big{\\{}\log\Big{(}\frac{c_{1}^{\prime}(\varepsilon_{1},\sigma_{1},\eta)}{(\kappa
C_{1})^{2}}\Big{)},0\Big{\\}}+\log{\mathscr{L}}_{0}\Big{)}\leq\eta.$ (3.48)

####  3.4.1. “Medium” case:

Assume $\frac{\lambda}{c_{4}}\leq(1-\beta_{1}){\mathscr{L}}<\frac{1}{R_{0}}$.
Lemma 9(i) yields

$\phi_{\theta}((1-\beta_{1})c_{4}{\mathscr{L}})c_{4}^{2}-c_{11}((1-\beta_{1}){\mathscr{L}})^{2c_{4}c_{8}}\geq\phi_{\theta}(\lambda)c_{4}^{2}-c_{11}\big{(}(1-\beta_{1}){\mathscr{L}}\big{)}^{2c_{4}c_{8}}.$
(3.49)

Hence, Proposition 15 gives

${\mathscr{L}}^{-2}\frac{|G|}{|C|}\mathcal{S}_{C}\geq\phi_{\theta}(\lambda)c_{4}^{2}-c_{11}R_{0}^{-2c_{4}c_{8}}-\mathcal{E}_{1}(c_{4}),$
(3.50)

where $c_{11}$ is defined in (3.45), and $\mathcal{E}_{1}(t)$ is defined by

$\mathcal{E}_{1}(t)=\frac{c_{6}}{{\mathscr{L}}_{0}}+{\mathscr{Q}}_{0}\alpha_{3}(\theta-1){\mathscr{L}}_{0}te^{-t{\mathscr{L}}_{0}}+\frac{4}{({\mathscr{L}}_{0}-\frac{1}{2})^{2}}e^{-t(2{\mathscr{L}}_{0}-1)}+\frac{c_{5}}{{\mathscr{L}}_{0}}e^{-2t{\mathscr{L}}_{0}}.$
(3.51)

####  3.4.2. “Small” case:

Assume
$\frac{\mu}{c_{4}{\mathscr{L}}^{\nu-1}}\leq(1-\beta_{1}){\mathscr{L}}\leq\frac{\lambda}{c_{4}}$.
As $(1-\beta_{1}){\mathscr{L}}\leq\frac{\lambda}{c_{4}}$ and $2c_{4}c_{8}>1$,
Lemma 9(ii) yields

$\phi_{\theta}((1-\beta_{1})c_{4}{\mathscr{L}})c_{4}^{2}-c_{11}((1-\beta_{1}){\mathscr{L}})^{2c_{4}c_{8}}\geq
2(\theta-1)^{2}e^{-2\lambda}c_{4}^{3}(1-\beta_{1}){\mathscr{L}}-c_{11}\Big{(}\frac{\lambda}{c_{4}}\Big{)}^{2c_{4}c_{8}-1}((1-\beta_{1}){\mathscr{L}}).$

Hence, by Proposition 15, we have

$((1-\beta_{1}){\mathscr{L}})^{-1}{\mathscr{L}}^{-2}\frac{|G|}{|C|}\mathcal{S}_{C}\geq
2(\theta-1)^{2}e^{-2\lambda}c_{4}^{3}-c_{11}\Big{(}\frac{\lambda}{c_{4}}\Big{)}^{2c_{4}c_{8}-1}-c_{6}\frac{c_{4}{\mathscr{L}}_{0}^{\nu-2}}{\mu}-\mathcal{E}_{2}(c_{4}),$
(3.52)

with

$\mathcal{E}_{2}(t)=\Big{(}{\mathscr{Q}}_{0}\alpha_{3}(\theta-1){\mathscr{L}}_{0}^{2}c_{4}e^{-t{\mathscr{L}}_{0}}+\frac{4}{{\mathscr{L}}_{0}(1-\frac{1}{2{\mathscr{L}}_{0}})^{2}}e^{-t(2{\mathscr{L}}_{0}-1)}+c_{5}e^{-2t{\mathscr{L}}_{0}}\Big{)}\frac{t{\mathscr{L}}_{0}^{\nu-2}}{\mu}.$
(3.53)

####  3.4.3. “Very small” case:

Assume $\frac{(\kappa
C_{1})^{2}}{{\mathscr{L}}}\leq(1-\beta_{1}){\mathscr{L}}\leq\frac{\mu}{c_{4}{\mathscr{L}}^{\nu-1}}$.
Then Lemma 9(ii) yields

$\phi_{\theta}((1-\beta_{1})c_{4}{\mathscr{L}})c_{4}^{2}-c_{11}((1-\beta_{1}){\mathscr{L}})^{2c_{4}c_{8}}\geq
2(\theta-1)^{2}e^{-\frac{2\mu}{{\mathscr{L}}_{0}^{\nu-1}}}c_{4}^{3}(1-\beta_{1}){\mathscr{L}}\\\
-c_{11}\Big{(}\frac{\mu}{c_{4}{\mathscr{L}}_{0}^{\nu-1}}\Big{)}^{2c_{4}c_{8}-1}(1-\beta_{1}){\mathscr{L}},$

which, combined with Proposition 15, yields

$((1-\beta_{1}){\mathscr{L}})^{-1}{\mathscr{L}}^{-2}\frac{|G|}{|C|}\mathcal{S}_{C}\geq
2(\theta-1)^{2}e^{-\frac{2\mu}{{\mathscr{L}}_{0}^{\nu-1}}}c_{4}^{3}-c_{11}\Big{(}\frac{\mu}{c_{4}{\mathscr{L}}_{0}^{\nu-1}}\Big{)}^{2c_{4}c_{8}-1}-\frac{c_{6}}{(\kappa
C_{1})^{2}}-\mathcal{E}_{3}(c_{4}),$ (3.54)

with

$\mathcal{E}_{3}(t)=\Big{(}{\mathscr{Q}}_{0}\alpha_{3}(\theta-1){\mathscr{L}}_{0}^{2}te^{-t{\mathscr{L}}_{0}}+\frac{4}{{\mathscr{L}}_{0}(1-\frac{1}{2{\mathscr{L}}_{0}})^{2}}e^{-t(2{\mathscr{L}}_{0}-1)}+c_{5}e^{-2t{\mathscr{L}}_{0}}\Big{)}\frac{1}{(\kappa
C_{1})^{2}}.$ (3.55)

####  3.4.4. “Extremely small” case:

We assume $1-\beta_{1}<(\kappa C_{1})^{2}{\mathscr{L}}^{-2}$. Here we use the
weight as introduced by [LaMoOd79], and used by [AhKw19-1]:

$k(s)=x^{s^{2}+s},$

with $x=d_{L}^{c_{12}}\geq 10^{10}$.

Thus, here $B$ is given by

$B=5c_{12}.$

First, we have the following explicit inequality from combining [AhKw19-1,
(3.2), Proposition 4.7, line 9 of p. 1453, and line -1 of p. 1452]:

$\begin{split}\frac{|G|}{|C|}\sum_{\mathfrak{p}\in\mathcal{P}(C),N\mathfrak{p}\leq
x^{5}}(\log
N\mathfrak{p})\hat{k}(N\mathfrak{p})\geq&k(1)-k(\beta_{1})-\sum_{|\gamma|\leq
1}|k(\rho)|\\\ &-19.17x{\mathscr{L}}-1.8292{\mathscr{L}}-5.4568x(\log
x)^{\frac{1}{2}}n_{L}{\mathscr{L}},\end{split}$ (3.56)

where, as later, the sum on the right is over non-trivial zeros
$\rho=\beta+i\gamma\neq\beta_{1}$ of $\zeta_{L}(s)$.333The values appearing in
(3.56) are respectively called $c_{20},c_{15}^{\prime},\alpha_{4}$ in
[AhKw19-1].
We now emphasise how to improve the estimate for $k(1)-k(\beta_{1})$ as well
as the contribution of the low-lying zeros of $\zeta_{L}(s)$. As $\beta_{1}$
is the closest to $1$, we use the mean value theorem to deduce

$k(1)-k(\beta_{1})\geq(1-\beta_{1})(2\beta_{1}+1)(\log
x)x^{\beta_{1}^{2}+\beta_{1}}=c_{12}(1-\beta_{1}){\mathscr{L}}(2\beta_{1}+1)e^{2c_{12}{\mathscr{L}}}e^{c_{12}(\beta_{1}^{2}+\beta_{1}-2){\mathscr{L}}}.$

Since

$2\beta_{1}+1\geq 3-\frac{2(\kappa C_{1})^{2}}{{\mathscr{L}}_{0}^{2}}\
\text{and}\
\beta_{1}^{2}+\beta_{1}-2=(\beta_{1}+2)(\beta_{1}-1)\geq-\frac{3(\kappa
C_{1})^{2}}{{\mathscr{L}}^{2}},$

we have

$k(1)-k(\beta_{1})\geq\phi_{0}(c_{12})\Big{(}{\mathscr{L}}(1-\beta_{1})e^{2c_{12}{\mathscr{L}}}\Big{)}c_{12},$
(3.57)

where

$\phi_{0}(t)=e^{-\frac{3(\kappa
C_{1})^{2}}{{\mathscr{L}}_{0}}t}\Big{(}3-\frac{2(\kappa
C_{1})^{2}}{{\mathscr{L}}_{0}^{2}}\Big{)}.$ (3.58)

We split the sum over the zeros, $\rho=\beta+i\gamma\neq\beta_{1}$, satisfying
$|\gamma|\leq 1$ and $\beta=1/2$: $\Sigma=\Sigma_{1}+\Sigma_{2}$, with

$\sideset{}{{}_{1}}{\sum}=\sum_{\beta<1/2}+\frac{1}{2}\sum_{\beta=1/2}\
\text{and}\
\sideset{}{{}_{2}}{\sum}=\frac{1}{2}\sum_{\beta=1/2}+\sum_{\beta>1/2}.$

We use the bound $|k(\rho)|\leq x^{\beta^{2}+\beta}\leq x^{3/4}$ in the first
part, together with the bound $N_{L}(1)\leq c_{13}{\mathscr{L}}$, as given in
(3.24), where

$c_{13}=\frac{1}{\pi}+0.296+\mathscr{Q}_{0}\Big{(}3.971-\frac{\log(2\pi
e)}{\pi}\Big{)}+\frac{3.969}{{\mathscr{L}}_{0}}.$ (3.59)

Thus,
$\Sigma_{1}\leq\frac{N_{L}(1)}{2}x^{\frac{3}{4}}\leq\frac{c_{13}}{2}{\mathscr{L}}x^{\frac{3}{4}}$.
Now, we assume $c_{12}>\frac{4}{5}c_{3}$ and deduce

$\sideset{}{{}_{1}}{\sum}\leq\frac{c_{13}}{2}e^{(c_{3}-\frac{5}{4}c_{12}){\mathscr{L}}}\Big{(}(1-\beta_{1}){\mathscr{L}}e^{2c_{12}{\mathscr{L}}}\Big{)}\leq\frac{c_{13}}{2}e^{-(\frac{5}{4}c_{12}-c_{3}){\mathscr{L}}_{0}}\Big{(}(1-\beta_{1}){\mathscr{L}}e^{2c_{12}{\mathscr{L}}}\Big{)}$
(3.60)

since
${\mathscr{L}}x^{\frac{3}{4}}=\big{(}x^{2}{\mathscr{L}}(1-\beta_{1})\big{)}\frac{x^{-5/4}}{(1-\beta_{1})}$,
where $x=e^{c_{12}{\mathscr{L}}}$, and $1-\beta_{1}\geq
d_{L}^{-c_{3}}=e^{-c_{3}{\mathscr{L}}}$ according to Theorem 4.
For $\Sigma_{2}$, we apply $\beta^{2}\leq 2\beta=2-2(1-\beta)$ so that

$|k(\rho)|\leq\frac{x^{-2(1-\beta)}}{(1-\beta_{1}){\mathscr{L}}}\Big{(}(1-\beta_{1}){\mathscr{L}}x^{2}\Big{)}=\frac{e^{-2c_{12}(1-\beta){\mathscr{L}}}}{(1-\beta_{1}){\mathscr{L}}}\Big{(}(1-\beta_{1}){\mathscr{L}}e^{2c_{12}{\mathscr{L}}}\Big{)}.$
(3.61)

Together with Deuring-Heilbronn phenomenon, as described in Theorem 2:

$(1-\beta){\mathscr{L}}\geq C_{2}\log
C_{1}-C_{2}\log\big{(}(1-\beta_{1}){\mathscr{L}}\big{)},$

we obtain

$e^{-2c_{12}(1-\beta){\mathscr{L}}}\leq
C_{1}^{-2c_{12}C_{2}}((1-\beta_{1}){\mathscr{L}})^{2c_{12}C_{2}},$ (3.62)

where $C_{1}$ and $C_{2}$ are defined in (2.28). Combining (3.61) and (3.62)
with $(1-\beta_{1}){\mathscr{L}}<(\kappa C_{1})^{2}{\mathscr{L}}^{-1}$,

$|k(\rho)|\leq\frac{C_{1}^{2c_{12}C_{2}-2}\kappa^{4c_{12}C_{2}-2}}{{\mathscr{L}}^{2c_{12}C_{2}-2}}\Big{(}(1-\beta_{1}){\mathscr{L}}e^{2c_{12}{\mathscr{L}}}\Big{)}.$
(3.63)

Thus, we have

$\sideset{}{{}_{2}}{\sum}\leq\frac{c_{13}}{2}\kappa^{2}\Big{(}\frac{\kappa^{2}C_{1}}{{\mathscr{L}}_{0}}\Big{)}^{2c_{12}C_{2}-2}\Big{(}(1-\beta_{1}){\mathscr{L}}e^{2c_{12}{\mathscr{L}}}\Big{)},$
(3.64)

as long as $c_{12}>\frac{1}{C_{2}}$. Combining (3.56), (3.57), (3.60), and
(3.64) finally gives that for

$c_{12}>\max\Big{(}\frac{1}{C_{2}},c_{3}\Big{)},$
$\Big{(}(1-\beta_{1}){\mathscr{L}}e^{2c_{12}{\mathscr{L}}}\Big{)}^{-1}\frac{|G|}{|C|}\sum_{\mathfrak{p}\in\mathcal{P}(C),N\mathfrak{p}\leq
d_{L}^{5c_{12}}}(\log N\mathfrak{p})\hat{k}(N\mathfrak{p})\\\
\geq\phi_{0}(c_{12})c_{12}-\frac{c_{13}\kappa^{2}}{2}\Big{(}\frac{\kappa^{2}C_{1}}{{\mathscr{L}}_{0}}\Big{)}^{2c_{12}C_{2}-2}-\mathcal{E}_{4}(c_{12}),$
(3.65)

where

$\mathcal{E}_{4}(t)=\frac{c_{13}}{2}e^{-(\frac{5}{4}t-c_{3}){\mathscr{L}}_{0}}+\Big{(}19.17+5.4568{\mathscr{Q}}_{0}{\mathscr{L}}_{0}^{\frac{3}{2}}t^{1/2}\Big{)}e^{-(t-c_{3}){\mathscr{L}}_{0}}+1.8292e^{-(2t-c_{3}){\mathscr{L}}_{0}}.$
(3.66)

###  3.5. Numerical results

We choose $\varepsilon_{1},\sigma_{1}$, and $\eta$ to make $c_{4}$ as small as
possible while (3.50), (3.52), (3.54), and (3.65) are satisfied.
Note that the choice for $\nu$ and $\eta$ are balanced in the inequalities
(3.52) and (3.54) as $\frac{c_{4}{\mathscr{L}}_{0}^{\nu-2}}{\mu}$ and
$\frac{\mu}{c_{4}{\mathscr{L}}_{0}^{\nu-1}}$ both have to be small enough.
Similarly, the choice of $\kappa$ is balanced by the inequalities (3.54) and
(3.65) as both $\frac{c_{6}}{(\kappa C_{1})^{2}}$ and
$\frac{\kappa^{2}C_{1}}{{\mathscr{L}}_{0}}$ need to be small. The choice for
$\varepsilon_{2}$ and $\sigma_{2}$ is to make $\frac{c_{6}}{(\kappa
C_{1})^{2}}$ as small as possible in (3.54). That choice was satisfactory to
keep (3.65) valid. Finally, we choose $\theta$ in each case to make $B$ as
small as possible. We recall that

$R_{0}=20\ \text{ and }\ R_{1}=1.24.$

We detail the case when $n_{0}=9$. From Table 3, for ${\mathscr{L}}\leq
2.29\cdot 10^{7}$, we have

$B\leq 1.7712.$

In the case where ${\mathscr{L}}>2.29\cdot 10^{7}$, we first investigate the
case where there are no exceptional zero. Choosing $\theta=12.83$ and
$\alpha=2.56$ and solving in $c_{4}$ so that (3.34) is satisfied then lead to

$B=10.4410.$

In the case the exceptional zero $\beta_{1}$ exists, we set

$\varepsilon_{1}=5.57,\ \sigma_{1}=4.45,\ \eta=0.025$

so that

$c_{1}^{\prime}(\varepsilon_{1},\sigma_{1},\eta)=0.002509182,\ \text{and}\
c_{2}^{\prime}(\varepsilon_{1},\sigma_{1},\eta)=0.04890427.$

This gives

$c_{8}=0.003324331,\ \text{and}\ c_{4}=150.4072.$

We also take

$\varepsilon_{2}=5.97,\ \sigma_{2}=4.5,\ \kappa=23,\ \lambda=0.2,\ \mu=0.1,\
\text{and}\ \nu=1.15,$

i.e., we work over the ranges:

$\frac{0.003330581}{{\mathscr{L}}}<\frac{0.0006648618}{{\mathscr{L}}^{0.15}}<0.001329724<0.05.$

We find in the “extremely small” case that

$c_{3}=9.85\ldots,\ c_{12}=34.97\ldots,\text{and}\ B=5c_{12}=174.8780.$

Finally, we choose $\alpha$ and $\theta$ in each other case and obtain

Case | Medium | Small | Very small
---|---|---|---
$\theta$ | $1.02$ | $1.02$ | $1.029$
$\alpha$ | $5.85$ | $0.17$ | $0.67$
$B$ | $306.8307$ | $306.8307$ | $309.5380$

For all the remaining degrees $n_{L}\neq 9$, we then do the calculations using
the same parameter values except for $\varepsilon_{2}$ which we choose
optimally for each degree.

Table 2. Table of admissible $(n_{0},d_{0},B)$ $n_{0}$ | 2 | 3 | 4 | 5 | 6
---|---|---|---|---|---
$d_{0}$ | $10^{10}$ | $10^{10}$ | $10^{8}$ | $10^{8}$ | $10^{8}$
$B$ | 223.2 | 231.7 | 249.1 | 259.8 | 271.1
$n_{0}$ | 7 | 8 | 9 | 10 | 11
$d_{0}$ | $10^{8}$ | $10^{7}$ | $2.29\cdot 10^{7}$ | $1.56\cdot 10^{8}$ | $3.91\cdot 10^{9}$
$B$ | 280.5 | 303.3 | 309.6 | 309.4 | 303.0
$n_{0}$ | 12 | 13 | 14 | 15 | 16
$d_{0}$ | $2.74\cdot 10^{10}$ | $7.56\cdot 10^{11}$ | $5.43\cdot 10^{12}$ | $1.61\cdot 10^{14}$ | $1.17\cdot 10^{15}$
$B$ | 303.2 | 298.4 | 298.8 | 295.1 | 295.6
$n_{0}$ | 17 | 18 | 19 | 20 | $21+$
$d_{0}$ | $3.70\cdot 10^{16}$ | $2.73\cdot 10^{17}$ | $9.03\cdot 10^{18}$ | $6.74\cdot 10^{19}$ | $10^{n_{L}}$
$B$ | 292.5 | 293.0 | 290.4 | 291.0 | $290.2$

We find all admissible values for $B$ are no larger than the above $309.5380$
(see Table 2 for the values of $B$ for each case). This proves that for any
Galois extension $L/K$ of number fields with Galois group $G$, $C$ a conjugacy
class in $G$, $n_{L}=n_{0}$ and $d_{L}\geq d_{0}$, there exists an unramified
prime $\mathfrak{p}$ of $K$, of degree one, such that
$\sigma_{\mathfrak{p}}=C$ and $N\mathfrak{p}\leq d_{L}^{B}$ with admissible
$(n_{0},d_{0},B)$ recorded in Table 2. Together with Fiori’s numerical
verifications recorded in Table 3, this concludes the proof of Theorem 1.

###  3.6. A summary of key ideas

We list here what we have done differently from [AhKw19-1].

(i) For $d_{L}$ relatively small, the least prime is bounded numerically (see
Appendix).

(ii) If the exceptional zeros exists, we have an enlarged region that contains
no non-exceptional (Proposition 6). (Cf. [AhKw19-1, Eq. (6.1)].)

(iii) We obtain a better version of the Deuring-Heilbronn phenomenon (cf.
[AhKw19-1, Theorem 7.3]) which is due to

  * •

the refined Turán’s power sum method established in [KaNgWo19] (cf. [LaMoOd79,
Theorem 4.2]), and

  * •

better estimates for $\frac{\Gamma^{\prime}}{\Gamma}$, i.e., Lemma 8 and
(2.25), (cf. [AhKw19-1, Lemmata 5.3 and 5.4]).

In addition, we do not follow [AhKw19-1, Theorem 7.3]) to split our
consideration into imaginary and non-imaginary cases but only consider the
non-trivial zeros of $\zeta_{L}(s)$ (as the Deuring-Heilbronn phenomenon is
“trivial” for the trivial zeros of $\zeta_{L}(s)$).

(iv) Regarding the weight $k$:

  * •

we change the weight $k$ as in (3.1) for the “non-extremely-small” cases.

  * •

we use better estimates for $k(1)-k(\beta_{1})$, namely, (3.5) (together with
Lemma 9) and (3.57).

(iv) Regarding the position of the exceptional zero $\beta_{1}$ of the
Dedekind zeta function:

  * •

we consider a more refined splitting for the position of $\beta_{1}$ (Section
3.4; cf. [AhKw19-1, Sec. 8]); our “small” case (when
$(1-\beta_{1}){\mathscr{L}}$ is smaller than constant size) is giving the
worst $B$. Splitting this further with a “very small” case allowed further
improvement.

(v) Regarding the zeros of the Dedekind zeta function:

  * •

we use the improved zero-density estimates (3.24), established in [HaShWo],
and (3.25) (cf. [AhKw19-1, Propositions 5.5 and 5.6]).

  * •

we obtain a better estimate for zeros $\rho$ with $|{\mathfrak{Im}}(\rho)|\leq
1$ (see Propositions 13 and 15; cf. [AhKw19-1, Lemma 8.2]) by

    * –

noticing that there are no non-exceptional zeros close to 1 (see (3.31)), and

    * –

balancing the use of zero-free region and zero repulsion in the comparison of
(a) and (b) done in the proof of Proposition 15 (cf. [AhKw19-1, (i) and (ii)
of the proof of Lemma 8.2]).

  * •

a consequence of not over-counting low-lying zeros is to improve the value of
$B$ in the “medium” case.

  * •

we realize that for the “extremely small” case the contribution of the zeros
$\rho$, with ${\mathfrak{Re}}(\rho)\leq\frac{1}{2}$, to $\sum_{\rho}|k(\rho)|$
is small (see (3.60); cf. [AhKw19-1, Lemma 8.5]). By this observation, we
essentially half the size of the sum.

  * •

we use the stronger repulsion of zeros $\rho$, with
$|{\mathfrak{Im}}(\rho)|\leq 1$, from Theorem 2 instead of the general one as
used in [AhKw19-1, Theorem 7.3(2)].

Lastly, as we used a different labeling for the constants from [AhKw19-1], we
provide here the following table for comparison:

Ahn-Kwon’s notation | $c_{7}$ | $c_{8}$ | $c_{10}$ | $c_{11}$ | $c_{12}$ | $c_{13}$ | $c_{14}$ | $c_{15}$ | $c_{16}$ | $c_{21}$ | $c_{23}$ | $\check{c}$
---|---|---|---|---|---|---|---|---|---|---|---|---
our new notation | $c_{1}$ | $c_{2}$ | $c_{3}$ | $c_{9}$ | $c_{8}=c_{10}$ | $c_{6}$ | $c_{11}$ | $c_{5}$ | $c_{4}$ | $c_{13}$ | $c_{12}$ | $8+\varepsilon$

Note that in [AhKw19-1], there is no constant corresponding to our $c_{7}$ as
the non-exceptional case is not treated separately there.

##  4\. Lower bound for the Chebotarev density theorem

###### Proof of Theorem 5.

Throughout this section, we shall adapt the notation used in [AhKw19-2]. We
let $c_{6}=11.7$ and $1-\beta_{1}\geq d_{L}^{-c_{6}}$. 444$c_{6}$ in this
section is our $c_{3}$ from Theorem 4. We shall assume $\log x\geq
d_{L}^{c_{6}}$. Let $a\in(1,2]$. We consider the kernel $k_{a}(s)$ defined by

$k_{a}(s)=\frac{(x^{s}-1)(a^{s}-1)}{s^{2}\log a}.$

Recall that the inverse Mellin transform of $k_{a}(s)$ is

$\widehat{k_{a}}(u)=\frac{1}{2\pi
i}\int_{2-i\infty}^{2+i\infty}k_{a}(s)u^{-s}ds=\left.\begin{cases}\frac{\log
u}{\log a}&\text{if }1<u<a,\\\ 1&\text{if }a\leq u\leq x,\\\ \frac{1}{\log
a}\log(\frac{ax}{u})&\text{if }x<u\leq ax,\\\ 0&\text{if }u>ax.\\\
\end{cases}\right.$

By [AhKw19-2, Lemma 2.3], we know that

$\mathcal{D}_{C}=\sum_{\mathfrak{p}\in\mathcal{P}(C)}(\log
N\mathfrak{p})\widehat{k_{a}}(N\mathfrak{p})\leq\tilde{\pi}_{C}(ax)\log x.$
(4.1)

Let $x_{0}=3^{c_{6}}$. Then for $x\geq\exp(x_{0})$, one has

$\frac{|G|}{|C|}\mathcal{D}_{C}\geq c_{43}(a)x,$ (4.2)

where

$c_{43}(a)=0.49\frac{a-1}{\log
a}-c_{41}e^{-c_{39}\sqrt{\frac{x_{0}}{2}}}-\Big{(}c_{35}x_{0}\log
x_{0}+c_{40}x_{0}\Big{)}e^{-\sqrt{\frac{x_{0}}{2}}}$

and $c_{35}$, $c_{39}$, $c_{40}$, and $c_{41}$ are constants, depending on
$x_{0}$ and $a$, which are defined in [AhKw19-1, pp. 304–306].

By (4.1) and (4.2), we have

$\frac{|G|}{|C|}\tilde{\pi}_{C}(ax)\log x\geq c_{43}(a)x.$

Hence, replacing $x$ by $x/a$, together with a simplification, we deduce

$\tilde{\pi}_{C}(x)\geq\frac{c_{43}(a)}{a}\frac{|C|}{|G|}\frac{x}{\log x-\log
a}\geq\frac{c_{43}(a)}{a}\frac{|C|}{|G|}\frac{x}{\log x}.$

Finally, we conclude the proof by choosing $a$ as close to $1$ as possible.
Namely, for $a=1.0001$, we find $m=\frac{c_{43}(a)}{a}=0.489975\ldots$. ∎

######  Remark.

Note that, for $a\approx 1$, $c_{25}=2(50.313...)\frac{a+1}{\log a}$,
$c_{35}\approx\frac{2}{c_{6}\log 3}c_{8}$,
$c_{39}\approx\frac{1}{\sqrt{29.57}}$, $c_{40}\approx\sqrt{a}\approx 1$, and
$c_{41}\approx\frac{5.7868c_{39}}{c_{6}}\frac{a+1}{\log a}$. Note that, as
$\frac{a-1}{\log a}\approx 1$ when $a\approx 1$, then
$m=\frac{c_{43}(a)}{a}\approx 0.49$. This value $0.49$ comes from the choice
of the kernel $k_{a}(s)$, so this may be improved by using a different kernel.
Note that for $a=2$ we find $m=\frac{c_{43}(a)}{a}=0.353460\ldots$.

##  Appendix A Numerical Verification of the Least Prime in the Chebotarev
Density Theorem - by Andrew Fiori555The majority of computations for this work
were provided on systems supported by Compute Canada. Additional hardware at
the University of Lethbridge was purchased through an NSERC RTI grant and
supported by funding from the University of Lethbridge.

###  A.1. Introduction

The purpose of this appendix is to document partial numerical verification of
different bounds for the least prime in the Chebotarev density theorem. There
are several known asymptotic bounds for the least prime [BaSo96, LaMoOd79,
Zam17, AhKw19-1, KaNgWo19], the shape of these bounds depends on whether one
assumes the GRH. For each of these bound shapes we document the worst case
behaviour for fields up to some bound on the discriminant and the Galois type
of the field. Here Galois type refers to the Galois closure of the field.

The numerical verification documented here involved two essential steps:

  1. (1)

Obtaining proven complete tabulations of number fields up to some discriminant
bound. We describe our methods in Section A.2.

  2. (2)

For each field, and each automorphism of that field, finding the least prime
in the Chebotarev density theorem. We describe our methods in Section A.3.

A summary of our results by degree is in Section A.4. More complete tables of
results are available from [FioWeb1]. Each step of this process made extensive
use of the PARI-GP library [PARI]

###  A.2. Tabulation of Number Fields

Many researchers have made contributions towards the development of
tabulations of number fields with small discriminant. In particular, we refer
the reader to the following (incomplete) list of references: [JoRo14, KlMa01,
Sim98, Hun57, Poh82, Mar85, Bel97, CoDyDOl03, ScPoDyD94, Oli90, BeMaOl90,
Oli92, DyD84, DyD87, PoMaDyD90, Sel99, DJ10, Sel01, DrJo09, Voi08]. Moreover,
several efforts have been made to construct reference databases particularly
by the PARI group, Kluners and Malle, and Jones and Roberts.

The above work and databases form the basis for our own tabulations which in
some cases extend these. The tabulations we have made are available in
[FioWeb2]. Note that extending these tabulations is an ongoing project.

####  A.2.1. Methods for Tabulations of Number Fields

For degree two fields tabulations are essentially trivial, and for degree
three fields the work of Belabas [Bel97] provides an efficient method to
obtain tabulations. For higher degree fields there are several strategies
which may be applicable depending on the type of the field.

  * •

For large degrees the Minkowski bounds provide strong lower bounds for the
minimal root discriminant of a field, see for example [DyD80] We used this as
the basis for completion results for degrees 9 and higher. Note that for a
field of degree $n_{L}$ which is $21$ or larger we have $d_{L}>10^{n_{L}}$.

  * •

If the type of the field being considered is primitive then a search based on
the geometry of numbers using ideas from [Hun57, Poh82] provides a method to
obtain a complete tabulation up to a chosen bound on the discriminant. We used
this method to extend the tabulations for relevant field types and in
particular to improve upon existing tabulations in degrees $4,5,6$, and $8$.

  * •

If the type of the field being tabulated admits an automorphism one may use
Kummer theory to build all possible relative extensions based on complete
tabulations of the relevant lower degree field. PARI-GP includes complete
implementations of the aspects of class field theory and Kummer theory
necessary to implement these methods. We used this method to extend the
tabulations for relevant field types in particular to improve the completion
in degree $4,6,8$.

  * •

If the type of field being tabulated admits a subfield, over which the
relative Galois group (of the relative Galois closure) is solvable, one can
use Kummer theory in a more elaborate way building off the ideas used in
[CoDyDOl03]. Such strategies also work if the Galois group over $\mathbb{Q}$
is solvable. PARI-GP includes complete implementations of the aspects of class
field theory and Kummer theory necessary to implement these methods. We used
this method to extend the tabulations for relevant field types in particular
to improve the completion in degree $6,9$.

  * •

If the field being tabulated admits a subfield then the geometry of numbers
methods of [Mar85] may be used to build all relative extensions. This method
may be used even when Kummer theory can be used. Kummer theory will typically
be simpler. We have not made use these methods in our search as none of the
extension degrees for which we can obtain results would benefit.

###  A.3. Verification of Least Prime

We now document our method of verification.

Suppose $L/\mathbb{Q}$ is a fixed field with absolute discriminant $d_{L}$.
Let $G={\rm Aut}_{\mathbb{Q}}(L)$ and $\sigma\in G$.

What we verify is that there exists a small prime $p$ of $\mathbb{Q}$ which is
unramified in $L$, for which there exists a prime $\mathcal{P}$ of $L$ over
$p$ such that

  1. (1)

$\sigma(\mathcal{P})=\mathcal{P}$.

  2. (2)

$\forall x\in\mathcal{O}_{L},\sigma(x)\equiv x^{p}\pmod{\mathcal{P}}$,

where small means relative to one of the types of bound under consideration
relative to the discriminant $d_{L}$ of $L$.

We note that this formulation differs from how one typically states the
conditions on the least prime in the Chebotarev density theorem. We state the
following claims without proof:

######  Claim.

If $K=L^{\sigma}$ is the fixed field of $\sigma$ then
$\mathfrak{p}=\mathcal{P}\cap K$ satisfies the usual conditions of the least
prime in the Chebotarev density theorem if and only if $\mathcal{P}$ satisfies
these slightly different conditions. Hence, we may find the smallest value of
$N_{K/\mathbb{Q}}(\mathfrak{p})=p$ by finding the smallest value of $p$.

######  Claim.

If $K\subset L^{\sigma}$ is the fixed field of $\sigma$ then if $\mathcal{P}$
satisfies these slightly different conditions then
$\mathfrak{p}=\mathcal{P}\cap K$ satisfies the conditions of the least prime
in the Chebotarev density theorem with respect to the image of $\sigma$ in
${\rm Aut}_{K}(L)$. Hence, we may find an upper bound for the smallest value
of $N_{K/\mathbb{Q}}(\mathfrak{p})=p$ by finding the smallest value of $p$.

The above two claims are useful because they reduce the total number of cases
that needed to be checked, but also because they make clear that an explicit
check for any given field and any given automorphism is actually
computationally practical. Indeed, PARI-GP includes all the tools necessary to
make such a check, and hence find the smallest $p$.

######  Remark 16.

Because $p$ is expected to be small, the runtime of this search is typically
very small.

We remark also that PARI-GP has all the tools necessary to compute ${\rm
Aut}_{\mathbb{Q}}(L)$.

###  A.4. Summary of Results

Table 3 summarizes the results we have obtained.

More complete tables, as well as searchable lists of all results for all
fields referenced can be found at [FioWeb1].

There are essentially three types of bounds considered

  1. A

bound $d_{L}^{A}$ which is based on the GRH-unconditional results.

  2. B

bound $B(\log d_{L})^{2}$ which is based on the GRH-conditional results.

  3. C

bound $C(3e^{\gamma}/\pi)^{2}\frac{(\log d_{L})^{2}(\log(2\log\log
d_{L}))^{2}}{(\log\log d_{L})^{2}}$ which is based on an upper bound for the
worst known family of fields [Fio18].

For each row of the table and each type of bound, there are no fields $L$ with
$n_{L}=n_{0}$ and $d_{L}\leq d_{0}$, for which the least prime in the
Chebotarev density theorem exceeds the bounds. Additionally, for each row, and
each type of bound, there exists a field $L$, with $n_{L}=n_{0}$, which (up to
the precision given) realizes the given bound. In many cases these worst case
fields will have absolute discriminants larger than the verification height.
One can thus interpret each row as giving a lower bound on the upper bound for
the least prime in the Chebotarev theorem. The column $d_{min}$ documents a
lower bound for the absolute discriminant of fields of that degree, so that
for all $L$ with $n_{L}=n_{0}$ we have $d_{L}\geq d_{min}$. We recall that for
degrees 9+ there are in fact no fields whatsoever with $n_{L}=n_{0}$ and
$d_{L}\leq d_{0}$. For degree $21$ and higher this verification height, and
$d_{min}$ can be taken as at least $10^{[L:Q]}$.

Table 3. Table of worst case bounds $n_{0}=[L:\mathbb{Q}]$ | $d_{min}$ |  Verification Height $d_{0}$ |  A |  B |  C
---|---|---|---|---|---
2 | $3$ | $16$ |  1.7712 |  5.7997 |  136.0600
2 |  | $16<d_{L}\leq 10^{10}$ |  0.8071 |  1.5802 |  1.0389
3 | $23$ | $10^{10}$ |  0.6591 |  0.8803 |  0.5661
4 | $117$ | $10^{8}$ |  0.6098 |  0.9979 |  0.6823
5 | $1609$ | $10^{8}$ |  0.3559 |  0.6668 |  0.5297
6 | $9747$ | $10^{8}$ |  0.4335 |  0.6668 |  0.5297
7 | $184607$ | $10^{8}$ |  0.2490 |  0.2050 |  0.1890
8 | $1257728$ | $10^{7}$ |  0.2925 |  0.5047 |  0.5327
9 | $2.29\cdot 10^{7}$ | $2.29\cdot 10^{7}$ |  0.2183 |  0.1692 |  0.1658
10 | $1.56\cdot 10^{8}$ | $1.56\cdot 10^{8}$ |  0.2295 |  0.4319 |  0.5069
11 | $3.91\cdot 10^{9}$ | $3.91\cdot 10^{9}$ |  0.1228 |  0.1047 |  0.1272
12 | $2.74\cdot 10^{10}$ | $2.74\cdot 10^{10}$ |  0.2107 |  0.3269 |  0.3393
13 | $7.56\cdot 10^{11}$ | $7.56\cdot 10^{11}$ |  0.1365 |  0.0627 |  0.0690
14 | $5.43\cdot 10^{12}$ | $5.43\cdot 10^{12}$ |  0.1582 |  0.4006 |  0.5268
15 | $1.61\cdot 10^{14}$ | $1.61\cdot 10^{14}$ |  0.1206 |  0.0866 |  0.1201
16 | $1.17\cdot 10^{15}$ | $1.17\cdot 10^{15}$ |  0.1463 |  0.2769 |  0.3410
17 | $3.70\cdot 10^{16}$ | $3.70\cdot 10^{16}$ |  0.1069 |  0.0478 |  0.0689
18 | $2.73\cdot 10^{17}$ | $2.73\cdot 10^{17}$ |  0.1279 |  0.3371 |  0.4778
19 | $9.03\cdot 10^{18}$ | $9.03\cdot 10^{18}$ |  0.0909 |  0.0327 |  0.0484
20+ | $6.74\cdot 10^{19}$ | $6.74\cdot 10^{19}$ |  0.1230 |  0.3370 |  0.4741

## References

  * [AhKw14] J.-H. Ahn and S.-H. Kwon, Some explicit zero-free regions for Hecke L-functions, J. Number Theory 145 (2014), 433–473.
  * [AhKw19-1] by same author, An explicit upper bound for the least prime ideal in the Chebotarev density theorem, Ann. Inst. Fourier (Grenoble) 69 (2019), 1411–1458.
  * [AhKw19-2] by same author, Lower estimates for the prime ideal of degree one counting function in the Chebotarev density theorem, Acta Arith. 191 (2019), no. 3, 289–307.
  * [BaSo96] E. Bach and J. Sorenson, Explicit bounds for primes in residue classes, Math. Comp. 65 (1996), no. 216, 1717–1735.
  * [Bel97] K. Belabas, _A fast algorithm to compute cubic fields_ , Math. Comp. 66 (1997), no. 219, 1213–1237.
  * [BeMaOl90] A.-M. Bergé, J. Martinet, and M. Olivier, _The computation of sextic fields with a quadratic subfield_ , Math. Comp. 54 (1990), no. 190, 869–884.
  * [CoDyDOl03] H. Cohen, F. Diaz y Diaz, and M. Olivier, _Constructing complete tables of quartic fields using Kummer theory_ , Math. Comp. 72 (2003), no. 242, 941–951.
  * [Das21] S. Das, An explicit version of Chebotarev’s density theorem MSc Thesis, Dec. 2020, https://opus.uleth.ca/bitstream/handle/10133/5825/DAS_SOURABHASHIS_MSC_2020.pdf?sequence=3&isAllowed=y
  * [Deu35] M. Deuring, Über den Tschebotareffschen Dichtigkeitssatz, Math. Ann. 110 (1) (1935), 414–415.
  * [DyD80] F. Diaz y Diaz, _Tables minorant la racine $n$-ième du discriminant d’un corps de degré $n$_, Publications Mathématiques d’Orsay 80 [Mathematical Publications of Orsay 80], vol. 6, Université de Paris-Sud, Département de Mathématique, Orsay, 1980.
  * [DyD84] by same author, _Valeurs minima du discriminant pour certains types de corps de degré $7$_, Ann. Inst. Fourier (Grenoble) 34 (1984), no. 3, 29–38.
  * [DyD87] by same author, _Petits discriminants des corps de nombres totalement imaginaires de degré $8$_, J. Number Theory 25 (1987), no. 1, 34–52.
  * [DrJo09] E. D. Driver and J. W. Jones, _A targeted Martinet search_ , Math. Comp. 78 (2009), no. 266, 1109–1117.
  * [DJ10] by same author, _Minimum discriminants of imprimitive decic fields_ , Experiment. Math. 19 (2010), no. 4, 475–479.
  * [Fio18] A. Fiori, _Lower bounds for the least prime in Chebotarev_ , Algebra Number Theory 13 (2019), no. 9, 2199–2203.
  * [FioWeb1] by same author, _The first prime in Chebotarev_ , 2019.
http://www.cs.uleth.ca/ fiori/ResearchCode/ExplicitChebotarev/

  * [FioWeb2] by same author, _Tabulations of number fields of bounded discriminant by Galois type_ , 2019.
http://www.cs.uleth.ca/ fiori/ResearchCode/LowDiscFields/

  * [HaShWo] E. Hasanalizade, Q. Shen, and P.-J. Wong Counting Zeros of Dedekind Zeta Functions, arXiv:2102.04663.
  * [Hun57] J. Hunter, _The minimum discriminants of quintic fields_ , Proc. Glasgow Math. Assoc. 3 (1957), 57–67.
  * [JoRo14] J. W. Jones and D. P. Roberts, _A database of number fields_ , LMS J. Comput. Math. 17 (2014), no. 1, 595–618.
  * [Kad12] H. Kadiri, Explicit zero-free regions for Dedekind zeta functions, Int. J. Number Theory 8 (2012), no. 1, 125–147.
  * [KaNgWo19] H. Kadiri, N. Ng, and P.-J. Wong, The least prime ideal in the Chebotarev density theorem, Proceedings of the American Mathematical Society 147 (2019), 2289–2303.
  * [KlMa01] J. Klüners and G. Malle, _A database for field extensions of the rationals_ , LMS J. Comput. Math. 4 (2001), 182–196.
  * [LaMoOd79] J. C. Lagarias, H. L. Montgomery, and A. M. Odlyzko, A bound for the least prime ideal in the Chebotarev density theorem, Invent. Math. 54 (1979), 271–296.
  * [LaOd77] J. C. Lagarias and A. M. Odlyzko, Effective versions of the Chebotarev density theorem, Algebraic number fields: L-functions and Galois properties (Proc. Sympos., Univ. Durham, Durham, 1975), pp. 409–464. Academic Press, London, 1977.
  * [Lee] E. S. Lee, On an explicit zero-free region for the Dedekind zeta-function, J. Number Theory 224 (2021), 307–322.
  * [LMFDB] The LMFDB Collaboration, The L-functions and Modular Forms Database, available from http://www.lmfdb.org.
  * [Mar85] J. Martinet, _Methodes géométriques dans la recherche des petits discriminants_ , Séminaire de théorie des nombres, Paris 1983–84, Progr. Math., vol. 59, Birkhäuser Boston, Boston, MA, 1985, pp. 147–179.
  * [Mon94] H. L. Montgomery, _Ten lectures on the interface between analytic number theory and harmonic analysis_ , CBMS Regional Conference Series in Mathematics, vol. 84, 1994.
  * [Oli90] M. Olivier, _Corps sextiques primitifs_ , Ann. Inst. Fourier (Grenoble) 40 (1990), no. 4, 757–767.
  * [Oli92] by same author, _The computation of sextic fields with a cubic subfield and no quadratic subfield_ , Math. Comp. 58 (1992), no. 197, 419–432.
  * [PARI] The PARI Group, Univ. Bordeaux, _PARI/GP version 2.11.2_, 2019, available from http://pari.math.u-bordeaux.fr/.
  * [Pla16] D. J. Platt, Numerical computations concerning the GRH, Math. Comp. 85 (2016), no. 302, 3009–3027.
  * [Poh82] M. Pohst, _On the computation of number fields of small discriminants including the minimum discriminants of sixth degree fields_ , J. Number Theory 14 (1982), no. 1, 99–117.
  * [PoMaDyD90] M. Pohst, J. Martinet, and F. Diaz y Diaz, _The minimum discriminant of totally real octic fields_ , J. Number Theory 36 (1990), no. 2, 145–159.
  * [ScPoDyD94] A. Schwarz, M. Pohst, and F. Diaz y Diaz, _A table of quintic number fields_ , Math. Comp. 63 (1994), no. 207, 361–376.
  * [Sel99] S. Selmane, _Non-primitive number fields of degree eight and of signature $(2,3)$, $(4,2)$ and $(6,1)$ with small discriminant_, Math. Comp. 68 (1999), no. 225, 333–344.
  * [Sel01] by same author, _Quadratic extensions of totally real quintic fields_ , Math. Comp. 70 (2001), no. 234, 837–843.
  * [Sim98] D. Simon, _Equations dans les corps de nombres et discriminants minimaux_ , Ph.D. thesis, 1998, Thèse de doctorat dirigée par H. Cohen, Mathématiques pures Bordeaux 1 1998, p. 152 p.
  * [ThZa17] J. Thorner and A. Zaman, An explicit bound for the least prime ideal in the Chebotarev density theorem, Algebra Number Theory 11 (2017), 1135–1197.
  * [Tol97] E. Tollis, Zeros of Dedekind Zeta Functions in the Critical Strip, Math. Comp. 66 (1997), 1295–1321.
  * [Voi08] J. Voight, _Enumeration of totally real number fields of bounded root discriminant_ , Algorithmic number theory, Lecture Notes in Comput. Sci., vol. 5011, Springer, Berlin, 2008, pp. 268–281.
  * [Win13] B. Winckler, Théorème de Chebotarev effectif, arXiv:1311.5715.
  * [Zam17] A. Zaman, Bounding the least prime ideal in the Chebotarev density theorem, Funct. Approx. Comment. Math. 57 (2017), no. 1, 115–142.