abconsymbol=c

# An explicit version of Bombieri’s log-free density estimate and Sárközy’s
theorem for shifted primes

Jesse Thorner  Department of Mathematics, University of Illinois, Urbana, IL
61801, USA<EMAIL_ADDRESS>and  Asif Zaman  Department of
Mathematics, University of Toronto, 40 St. George Street, Toronto, ON, Canada
M5S 2E4<EMAIL_ADDRESS>
###### Abstract.

We make explicit Bombieri’s refinement of Gallagher’s log-free “large sieve
density estimate near $\sigma=1$” for Dirichlet $L$-functions. We use this
estimate and recent work of Green to prove that if $N\geq 2$ is an integer,
$A\subseteq\\{1,\ldots,N\\}$, and for all primes $p$ no two elements in $A$
differ by $p-1$, then $|A|\ll N^{1-1/10^{18}}$. This strengthens a theorem of
Sárközy.

##  1\. Introduction and statement of results

Let $N\geq 2$ be an integer, and let $\delta(N)$ be the relative density of a
largest set $A\subseteq\\{1,\ldots,N\\}$ such that for all primes $p$, no two
elements in $A$ differ by $p-1$. Erdős conjectured that $\delta(N)\to 0$ as
$N\to\infty$, and Sárközy [32] proved the conjecture in the form
$\delta(N)\ll(\log\log N)^{-2+o(1)}$. Several improvements followed (e.g.,
Lucier [20] and Ruzsa and Sanders [31]). Wang [35] proved that there exists an
absolute and effectively computable constant $\Cl[abcon]{Wang}>0$ such that
$\delta(N)\ll\exp(-\Cr{Wang}(\log N)^{1/3})$. The proofs in [31, 35] use the
distribution of zeros of Dirichlet $L$-functions. Assuming the generalized
Riemann hypothesis for Dirichlet $L$-functions (GRH), Ruzsa and Sanders [31]
proved that $\delta(N)\ll\exp(-\Cr{Wang}(\log N)^{1/2})$ and asked whether GRH
implies the existence of a power-saving bound on $\delta(N)$.

In a recent tour de force, Green [10] unconditionally proved that such a
power-saving bound exists—that is, there exists a constant
$\Cl[abcon]{Green}>0$ such that $\delta(N)\ll N^{-\Cr{Green}}$. Assuming GRH,
Green’s arguments simplify to show that any fixed $0<\Cr{Green}<\frac{1}{12}$
is admissible [9], but an unconditional admissible value of $\Cr{Green}$ was
not produced. In this paper, we produce such an admissible value.

######  Theorem 1.1.

If $A\subseteq\\{1,\ldots,N\\}$ and no prime $p$ satisfies $p-1\in A-A$, then
$|A|\ll N^{1-1/10^{18}}$.

The new ingredient that we supply in order to deduce Theorem 1.1 from the work
in [10] is a certain log-free zero density estimate for Dirichlet
$L$-functions. Such a result was first proved by Linnik as part of his proof
that there exists an absolute and effectively computable constant $b>0$ such
the least prime in an arithmetic progression modulo $q$ is at most $q^{b}$
[19]. For an integer $q\geq 1$, let $\chi\,(\mathrm{mod}\,\,q)$ be a Dirichlet
character, and let $L(s,\chi)$ be its $L$-function. For $Q\geq 3$ and
$\sigma\geq 0$, define

(1.1) $\mathscr{L}(s,Q):=\prod_{1\leq q\leq
Q}\prod_{\begin{subarray}{c}\chi\,(\mathrm{mod}\,\,q)\\\ \textup{$\chi$
primitive}\end{subarray}}L(s,\chi),\qquad\beta_{1}(Q):=\max\\{\beta\in\mathbb{R}\colon\mathscr{L}(\beta,Q)=0\\}$

and

(1.2) $\displaystyle N(\sigma,Q)$
$\displaystyle:=\\#\\{\rho=\beta+i\gamma\colon\beta>\sigma,~{}|\gamma|\leq
Q,~{}\mathscr{L}(\rho,Q)=0\\},$ $\displaystyle N^{*}(\sigma,Q)$
$\displaystyle:=\\#\\{\rho=\beta+i\gamma\neq\beta_{1}(Q)\colon\beta>\sigma,~{}|\gamma|\leq
Q,~{}\mathscr{L}(\rho,Q)=0\\}.$

(All sums and counts over $\rho$ weigh the contribution for each $\rho$ with
its multiplicity.) Gallagher [8] used Turán’s lower bound for power sums with
a “pre-sifted” large sieve inequality (Lemma 5.1) to prove that there exist
absolute and effectively computable constants $\Cl[abcon]{Gallagher1}>0$ and
$\Cl[abcon]{Gallagher2}>0$ such that

(1.3)
$N(\sigma,Q)\leq\Cr{Gallagher1}Q^{\Cr{Gallagher2}(1-\sigma)},\qquad\sigma\geq
0,\quad Q\geq 3.$

Jutila [16] used Selberg’s theory of pseudocharacters to sharpen (1.3),
proving that for all $\varepsilon>0$, there exists an effectively computable
constant $c(\varepsilon)>0$ such that

(1.4) $N(\sigma,Q)\leq
c(\varepsilon)Q^{(6+\varepsilon)(1-\sigma)},\qquad\sigma\geq 4/5,\quad Q\geq
3.$

The exponent in (1.4) is as strong as the generalized Lindelöf hypothesis
implies. Another refinement comes from Bombieri [3], who proved that there
exist absolute and effectively computable constants $\Cl[abcon]{Bombieri1}>0$
and $\Cl[abcon]{Bombieri2}>0$ such that

(1.5) $N^{*}(\sigma,Q)\leq\Cr{Bombieri1}(1-\beta_{1}(Q))(\log
Q)Q^{\Cr{Bombieri2}(1-\sigma)},\qquad\sigma\geq 0,\quad Q\geq 3.$

If $\beta_{1}(Q)$ and $\sigma$ are suitably close to $1$ as a function of $Q$,
then (1.5) improves upon (1.3) and (1.4).

We prove explicit versions of (1.3) and (1.5) that hold for all $Q\geq 3$ and
all $\sigma\geq\frac{39}{40}$ (see Corollary 6.1 below for versions that
permit $\sigma\geq 0$). These appear to be the first explicit log-free zero
density estimates for Dirichlet $L$-functions that do not require $Q$ to be
“sufficiently large” or $\sigma$ to be “sufficiently close” to $1$, where
sufficiency is not explicitly quantified. Such estimates are useful in
contexts beyond our proof of Theorem 1.1.

######  Theorem 1.2.

Let $Q\geq 3$ and $\sigma\geq\frac{39}{40}$. If $N(\sigma,Q)$ and
$N^{*}(\sigma,Q)$ are as in (1.2), then

$N(\sigma,Q)\leq 10^{88}(10^{421}Q^{99})^{1-\sigma},\qquad N^{*}(\sigma,Q)\leq
10^{93}\min\\{1,(1-\beta_{1}(Q))\log Q\\}(10^{466}Q^{170})^{1-\sigma}.$

######  Remark.

If $Q\geq 5$, then $\mathscr{L}(0,Q)=0$ and $\mathscr{L}(\sigma,Q)\neq 0$ for
$\sigma\geq 1$. Thus, we have that $\beta_{1}(Q)\in[0,1)$.

The proof of Theorem 1.1 is sensitive to the size of the implied constants, so
we cannot use (1.4) without making $c(\varepsilon)$ explicit first. We did not
try to estimate $c(\varepsilon)$ because this requires us to make certain
intricate results of Huxley [15] on large values of Dirichlet polynomials
completely explicit in all ranges. While we avoided this potentially onerous
task for the sake of brevity, we encourage the interested reader to make (1.4)
explicit for all $Q\geq 3$ and all $\sigma\geq\frac{4}{5}$. Our proof of
Theorem 1.1 shows that further improvements to Theorem 1.2 do not
substantially improve the power savings in Theorem 1.1 without a numerical
optimization of the arguments in [10] or a substantial numerical improvement
in the standard zero-free region for $\mathscr{L}(s,Q)$.

### Outline of the paper

Section 2 provides auxiliary results on the distribution of zeros of Dirichlet
$L$-functions. Section 3 contains notation and conventions and a discussion
the central optimization problem in (3.1) whose solution is one of two key
results (along with Corollary 2.12) that determines the quality of the
constants in Theorem 1.2. In Section 4, we prove our zero detection result for
zeros of Dirichlet $L$-functions (Theorem 4.2) that serves as the basis for
our proof of Theorem 1.2 in Section 5. We use Theorem 1.2 and the work in [10]
to prove Theorem 1.1 in Section 6.

### Acknowledgements

We thank Ben Green, Peter Humphries, Nathan Ng, Sarah Peluse, and Joni
Teräväinen for helpful conversations and the anonymous referee for helpful
comments. We performed all numerical calculations with Mathematica 12.

##  2\. Auxiliary results on Dirichlet $L$-functions

Let $\chi\,(\mathrm{mod}\,\,q_{\chi})$ be a primitive Dirichlet character, and
let

$L(s,\chi)=\prod_{p}(1-\chi(p)p^{-s})^{-1}=\sum_{n=1}^{\infty}\frac{\chi(n)}{n^{s}},\qquad\textup{Re}(s)>1$

be its $L$-function. We let $\delta_{\chi}$ be the indicator function for the
trivial character modulo 1, whose $L$-function is the Riemann zeta function
$\zeta(s)$.

###  2.1. The Hadamard factorization

All of the content in this subsection can be found in [6]. Let

$\mathfrak{a}_{\chi}=\frac{1-\chi(-1)}{2},\qquad\tau(\chi)=\sum_{j=1}^{q_{\chi}}\chi(n)e^{2\pi
ij/q_{\chi}}.$

Since $\chi$ is primitive, we have that $|\tau(\chi)|=\sqrt{q_{\chi}}$. The
completed $L$-function

$\Lambda(s,\chi)=(s(s-1))^{\delta_{\chi}}\Big{(}\frac{q_{\chi}}{\pi}\Big{)}^{\frac{s+\mathfrak{a}_{\chi}}{2}}\Gamma\Big{(}\frac{s+\mathfrak{a}_{\chi}}{2}\Big{)}L(s,\chi)$

is entire of order one and satisfies the functional equation

(2.1)
$\Lambda(s,\chi)=\frac{\tau(\chi)}{i^{\mathfrak{a}_{\chi}}\sqrt{q_{\chi}}}\Lambda(1-s,\overline{\chi}).$

Since $\Lambda(s,\chi)$ is entire of order one, it exhibits a Hadamard
factorization

$\Lambda(s,\chi)=e^{A_{\chi}+B_{\chi}s}\prod_{\Lambda(\rho,\chi)=0}\Big{(}1-\frac{s}{\rho}\Big{)}e^{\frac{s}{\rho}}.$

The zeros $\rho=\beta+i\gamma$ are the nontrivial zeros of $L(s,\chi)$, and
they satisfy $0\leq\beta\leq 1$ and $\gamma\in\mathbb{R}$. The poles of
$s^{\delta_{\chi}}\Gamma(\frac{s+\mathfrak{a}_{\chi}}{2})$ are the trivial
zeros; these form the sequence
$(-2(j+\delta_{\chi})-\mathfrak{a}_{\chi})_{j=0}^{\infty}$. Upon taking
logarithmic derivatives, we find that

(2.2)
$\frac{L^{\prime}}{L}(s,\chi)=B_{\chi}+\sum_{\rho}\Big{(}\frac{1}{s-\rho}+\frac{1}{\rho}\Big{)}-\delta_{\chi}\Big{(}\frac{1}{s-1}+\frac{1}{s}\Big{)}-\frac{1}{2}\log\frac{q_{\chi}}{\pi}-\frac{1}{2}\frac{\Gamma^{\prime}}{\Gamma}\Big{(}\frac{s+\mathfrak{a}_{\chi}}{2}\Big{)}.$

Since

$\textup{Re}(B_{\chi})=-\sum_{\rho}\textup{Re}\Big{(}\frac{1}{\rho}\Big{)},$

it follows that

(2.3)
$\textup{Re}\Big{(}\frac{L^{\prime}}{L}(s,\chi)\Big{)}=\sum_{\rho}\textup{Re}\Big{(}\frac{1}{s-\rho}\Big{)}-\delta_{\chi}\textup{Re}\Big{(}\frac{1}{s-1}+\frac{1}{s}\Big{)}-\frac{1}{2}\log\frac{q_{\chi}}{\pi}-\frac{1}{2}\textup{Re}\Big{(}\frac{\Gamma^{\prime}}{\Gamma}\Big{(}\frac{s+\mathfrak{a}_{\chi}}{2}\Big{)}\Big{)}.$

###  2.2. Zero-free regions

We assemble some results on zero-free regions for $L(s,\chi)$ (that together
will imply Theorem 2.6) along with a few other useful results.

######  Lemma 2.1.

If $\rho=\beta+i\gamma$ is a nontrivial zero of $\zeta(s)$ and $|\gamma|\leq
3\,000\,175\,332\,800$, then $\beta=\frac{1}{2}$. If
$|\gamma|>3\,000\,175\,332\,800$, then $\beta\leq
1-1/(5.573\,412\log|\gamma|)$.

###### Proof.

The first part is [25, Theorem 1]. The second part is [24, Theorem 1]. ∎

######  Lemma 2.2.

If $q\leq 400\,000$ and $\chi\,(\mathrm{mod}\,\,q)$ is a nontrivial primitive
Dirichlet character, then $L(s,\chi)$ has no real nontrivial zeros and does
not vanish in the region

$0<\Big{|}\textup{Re}(s)-\frac{1}{2}\Big{|}\leq\frac{1}{2},\qquad
0<|\textup{Im}(s)|\leq\max\Big{\\{}\frac{10^{8}}{q},\frac{37\,500\,000(2-\mathfrak{a}_{\chi})}{q}+200\Big{\\}}.$

###### Proof.

This follows from [26, Theorems 10.1 and 10.2]. ∎

######  Lemma 2.3.

Let $q\geq 3$. The product $\prod_{\chi\,(\mathrm{mod}\,\,q)}L(s,\chi)$ has at
most one zero in the region

$\textup{Re}(s)\geq
1-\frac{\Cl[abcon]{ZFR}}{\log\max\\{q,q|\textup{Im}(s)|\\}},\qquad\Cr{ZFR}=\frac{1}{9.645\,908\,801}.$

If such a zero $\rho_{1}$ exists, then $q>400\,000$, $\rho_{1}$ is a real and
simple zero of $\mathscr{L}_{q}(s)$, and there exists a unique quadratic
character $\chi_{1}\,(\mathrm{mod}\,\,q)$ such that $L(\rho_{1},\chi_{1})=0$.

###### Proof.

When $q>400\,000$, McCurley [22, Theorem 1] proved this with
$\max\\{q,q|\textup{Im}(s)|\\}$ replaced by
$\max\\{q,q|\textup{Im}(s)|,10\\}$. The claimed improvements follow from Lemma
2.2. ∎

######  Lemma 2.4.

Let $\chi\,(\mathrm{mod}\,\,q)$ and
$\chi^{\prime}\,(\mathrm{mod}\,\,q^{\prime})$ be distinct real primitive
characters with $\min\\{q,q^{\prime}\\}>400\,000$. If $\beta$ (resp.
$\beta^{\prime}$) is a real zero of $L(s,\chi)$ (resp. $L(s,\chi^{\prime})$),
then

$\min\\{\beta,\beta^{\prime}\\}<1-\frac{15-10\sqrt{2}}{(5-\sqrt{5})\log(\frac{q^{\prime}q}{17})}=1-\frac{0.310\,3782\ldots}{\log(\frac{q^{\prime}q}{17})}.$

###### Proof.

McCurley [22, Theorem 2] proved this with $\frac{q^{\prime}q}{17}$ replaced by
$\max\\{\frac{q^{\prime}q}{17},13\\}$. When $\min\\{q,q^{\prime}\\}\leq
400\,000$, Lemma 2.2 implies that at least one of $L(s,\chi)$ and
$L(s,\chi^{\prime})$ will have no real zero at all. This means that we may
restrict consideration to $q$ and $q^{\prime}$ such that
$\max\\{\frac{q^{\prime}q}{17},13\\}=\frac{q^{\prime}q}{17}$. ∎

######  Corollary 2.5 (Page).

Let $Q\geq 3$. The product

$\mathscr{M}(s,Q):=\prod_{q\leq
Q}\prod_{\begin{subarray}{c}\chi\,(\mathrm{mod}\,\,q)\\\ \textup{$\chi$
primitive}\\\ \textup{$\chi$ quadratic}\end{subarray}}L(s,\chi)$

has at most one real zero in the interval

$s\geq 1-\frac{15-10\sqrt{2}}{2(5-\sqrt{5})\log
Q}=1-\frac{0.155\,1891\ldots}{\log Q}.$

If such a real zero, say $\beta_{1}(Q)$, exists, then $Q>400\,000$,
$\beta_{1}(Q)$ is a simple zero of $\mathscr{M}(s,Q)$, and there exists
$q_{1}\in(400\,000,Q]$ and a primitive quadratic character
$\chi_{1}\,(\mathrm{mod}\,\,q_{1})$ such that $L(\beta_{1}(Q),\chi_{1})=0$.

###### Proof.

It suffices to assume that $\beta_{1}(Q)$ exists. By Lemma 2.2, this ensures
that $Q>400\,000$. Suppose to the contrary that there exist two primitive
quadratic characters $\chi\,(\mathrm{mod}\,\,q)$ and
$\chi^{\prime}\,(\mathrm{mod}\,\,q^{\prime})$ such that $q,q^{\prime}\leq Q$
and $L(\beta_{1}(Q),\chi)=L(\beta_{1}(Q),\chi^{\prime})=0$. By Lemma 2.2, we
have that $\min\\{q,q^{\prime}\\}>400\,000$. It follows that

$\beta_{1}(Q)\geq 1-\frac{15-10\sqrt{2}}{2(5-\sqrt{5})\log Q}\geq
1-\frac{15-10\sqrt{2}}{(5-\sqrt{5})\log(q^{\prime}q)},$

contradicting Lemma 2.4. Therefore, at most one such character exists. The
simplicity of $\beta_{1}(Q)$ follows from Lemma 2.3. ∎

######  Theorem 2.6.

Let $Q\geq 3$. Let $\mathscr{L}(s,Q)$ and $\beta_{1}(Q)$ be as in (1.1), and
let $\Cr{ZFR}$ be as in Lemma 2.3. Except possibly at $\beta_{1}(Q)$,
$\mathscr{L}(s,Q)$ is nonzero in the region

$\textup{Re}(s)\geq 1-\frac{\Cr{ZFR}}{\log\max\\{Q,Q|\textup{Im}(s)|\\}}.$

If $\beta_{1}(Q)\geq 1-\frac{\Cr{ZFR}}{\log Q}$, then $Q>400\,000$,
$\beta_{1}(Q)$ is a simple zero of $\mathscr{L}(s,Q)$, there exists one
primitive Dirichlet character $\chi_{1}\,(\mathrm{mod}\,\,q_{1})$ with
$q_{1}\in(400\,000,Q]$ such that $L(\beta_{1}(Q),\chi_{1})=0$, and
$\beta_{1}(Q)\leq 1-100/(\sqrt{q_{1}}(\log q_{1})^{2})$.

###### Proof.

The bound on $\beta_{1}(Q)$ is proved in [4, 5]. Lemma 2.3 and Corollary 2.5
imply the rest. ∎

###  2.3. Counting zeros

We use the following estimate for the $\Gamma$-function.

######  Lemma 2.7.

If $\textup{Re}(z)\geq\frac{1}{2}$, then

$\textup{Re}\Big{(}\frac{\Gamma^{\prime}}{\Gamma}\Big{(}\frac{z}{2}\Big{)}\Big{)}\leq\log|z|-\gamma_{\mathbb{Q}},\qquad\gamma_{\mathbb{Q}}:=\lim_{N\to\infty}\Big{(}\sum_{n=1}^{N}\frac{1}{n}-\log
N\Big{)}=0.5772156649\ldots$

###### Proof.

In [14, Lemma 4.1], it is proved that if
$\Gamma_{\mathbb{R}}(z)=\pi^{-z/2}\Gamma(\frac{z}{2})$ and
$\textup{Re}(z)\geq\frac{1}{2}$, then

$\textup{Re}\Big{(}\frac{\Gamma_{\mathbb{R}}^{\prime}}{\Gamma_{\mathbb{R}}}(z)\Big{)}\leq-\frac{\log\pi+\gamma_{\mathbb{Q}}}{2}+\frac{\log|z|}{2}.$

Since
$\frac{\Gamma_{\mathbb{R}}^{\prime}}{\Gamma_{\mathbb{R}}}(z)=\frac{1}{2}\frac{\Gamma^{\prime}}{\Gamma}(\frac{z}{2})-\frac{\log\pi}{2}$,
the lemma follows. ∎

Given $\sigma\geq 0$, $T\geq 0$, $Q\geq 1$, $1\leq q\leq Q$, and a primitive
Dirichlet character $\chi\,(\mathrm{mod}\,\,q)$, we define

(2.4)
$N_{\chi}(\sigma,T):=\\#\\{\rho=\beta+i\gamma\colon\beta\geq\sigma,~{}|\gamma|\leq
T,~{}L(\rho,\chi)=0\\}.$

It follows that $N(\sigma,Q)$ in (1.2) can be written as

(2.5) $N(\sigma,Q)=\sum_{q\leq
Q}~{}\sum_{\begin{subarray}{c}\chi\,(\mathrm{mod}\,\,q)\\\ \textup{$\chi$
primitive}\end{subarray}}N_{\chi}(\sigma,Q).$

######  Lemma 2.8.

If $Q\geq 10^{4}$, then $N(0,Q)\leq 0.64Q^{3}\log Q$.

###### Proof.

If $\chi\,(\mathrm{mod}\,\,q)$ is primitive, then by [2, Corollary 1.2] and
[29, Theorem 19], the quantity $N_{\chi}(\sigma,T)$ in (2.4) satisfies

$\Big{|}N_{\chi}(0,T)-\frac{T}{\pi}\log\frac{qT}{2\pi
e}\Big{|}\leq\begin{cases}\min\\{0.247\log(qT)+6.894,0.298\log(qT)+4.358\\}&\mbox{if
$\chi\neq 1$ and $T\geq\frac{5}{7}$,}\\\ 0.137\log T+0.443\log\log
T+2.463&\mbox{if $\chi=1$ and $T>14$.}\end{cases}$

By Lemma 2.2, if $0\leq T\leq 14$, then $N_{1}(0,T)=0$. By (2.5), these
estimates imply the lemma. ∎

Our proof of Theorem 1.2 relies crucially on our ability to efficiently count
zeros in circles of small radius that are centered near the line
$\textup{Re}(s)=1$.

######  Lemma 2.9.

Let $\sigma\geq 1$, $0<r\leq 1$, $q\geq 1$, $T\geq 1$, $t\in\mathbb{R}$,
$|t|\leq T$, and $\chi\,(\mathrm{mod}\,\,q)$ be a primitive Dirichlet
character. Define

(2.6) $n_{\chi}(r,\sigma+it):=\\#\\{\rho=\beta+i\gamma\colon
0<\beta<1,~{}L(\rho,\chi)=0,~{}|\sigma+it-\rho|\leq r\\}.$

The bound $n_{\chi}(r,\sigma+it)\leq n_{\chi}(r,1+it)\leq
n_{\chi}(2r,1+r+it)\leq r(2\log(qT)-1)+4$ holds.

###### Proof.

If $t\in\mathbb{R}$, $\sigma>1$, and $0<r\leq 1$, then it follows from the
geometry of complex numbers that

$\displaystyle n_{\chi}(r,\sigma+it)\leq n_{\chi}(r,1+it)\leq
n_{\chi}(2r,1+r+it)$ $\displaystyle\leq 4r\sum_{|1+r+it-\rho|\leq
2r}\textup{Re}\Big{(}\frac{1}{1+r+it-\rho}\Big{)}$ $\displaystyle\leq
4r\sum_{\rho}\textup{Re}\Big{(}\frac{1}{1+r+it-\rho}\Big{)}.$

By (2.3), the last line equals

$4r\Big{(}\textup{Re}\frac{L^{\prime}}{L}(1+r+it,\chi)+\frac{1}{2}\log\frac{q}{\pi}+\frac{1}{2}\textup{Re}\frac{\Gamma^{\prime}}{\Gamma}\Big{(}\frac{1+r+\mathfrak{a}_{\chi}+it}{2}\Big{)}+\delta_{\chi}\textup{Re}\Big{(}\frac{1}{1+r+it}+\frac{1}{r+it}\Big{)}\Big{)}.$

Note that

$\textup{Re}\Big{(}\frac{L^{\prime}}{L}(1+r+it,\chi)\Big{)}\leq-\frac{\zeta^{\prime}}{\zeta}(1+r)\leq\frac{1}{r},$

and Lemma 2.7 implies that (since $|t|\leq T$ and $T\geq 1$)

$\frac{1}{2}\textup{Re}\frac{\Gamma^{\prime}}{\Gamma}\Big{(}\frac{1+r+\mathfrak{a}_{\chi}+it}{2}\Big{)}\leq\frac{\frac{1}{2}\log((1+r+\mathfrak{a}_{\chi})^{2}+t^{2})-\gamma_{\mathbb{Q}}}{2}\leq\frac{\log
T+\frac{1}{2}\log 10-\gamma_{\mathbb{Q}}}{2}.$

Finally, if $\chi$ is trivial, then $L(s,\chi)=\zeta(s)$ does not vanish when
$0\leq\textup{Re}(s)\leq 1$ and $|\textup{Im}(s)|\leq 14.13$. Therefore, it
suffices to assume that $|t|\geq 13.1$, in which case we have that

$\delta_{\chi}\textup{Re}\Big{(}\frac{1}{1+r+it}+\frac{1}{r+it}\Big{)}\leq
0.017\,182\,3\delta_{\chi}.$

The lemma follows once we collect the estimates above. ∎

Following Heath-Brown [11], we will substantially improve Lemma 2.9 in a
certain range of $r$; the quality of our improvement depends on how well one
can explicitly bound $L(s,\chi)$ on the line $\textup{Re}(s)=\frac{1}{2}$ in
terms of $q$ and $|\textup{Im}(s)|$. We give a short and self-contained proof
of an explicit version of the convexity bound
$L(\frac{1}{2}+it,\chi)\ll(q(|t|+1))^{1/4}$ (without the usual $+\varepsilon$
in the exponent).

######  Proposition 2.10.

If $\chi\,(\mathrm{mod}\,\,q_{\chi})$ is primitive and $t\in\mathbb{R}$, then
$|L(\tfrac{1}{2}+it,\chi)|\leq 2.97655(q_{\chi}|1+it|)^{\frac{1}{4}}$.

######  Remark.

When $\chi\neq 1$, this improves [12, (2) and (3)] and [28, Lemma 2.4] for all
$q\geq 3$ and $t\in\mathbb{R}$.

###### Proof.

First, assume that $\chi\neq 1$. By applying [1, Lemma 3.1] with
$F(x)=L(x+it,\chi)$ evaluated at $x=\frac{1}{2}$, we have that

$\log|L(\tfrac{1}{2}+it,\chi)|\leq\frac{1}{2}\int_{\mathbb{R}}\log|L(i(t+v),\chi)L(1+i(t+v),\chi)|\frac{dv}{\cosh(\pi
v)}.$

Since $L(z,\chi)=\overline{L(\overline{z},\overline{\chi})}$, it follows from
(2.1) that

$|L(i(t+v),\chi)|=\sqrt{\frac{q_{\chi}}{\pi}}\Big{|}\frac{\Gamma(\frac{1+\mathfrak{a}_{\chi}+i(t+v)}{2})}{\Gamma(\frac{\mathfrak{a}_{\chi}-i(t+v)}{2})}L(1+i(t+v),\chi)\Big{|}.$

The bound

$\Big{|}\frac{\Gamma(\frac{1+\mathfrak{a}_{\chi}+i(t+v)}{2})}{\Gamma(\frac{\mathfrak{a}_{\chi}-i(t+v)}{2})}\Big{|}\leq\frac{|1+i(t+v)|^{\frac{1}{2}}}{\sqrt{2}}$

follows from the convexity bounds [27, Lemma 1] (when $\mathfrak{a}_{\chi}=0$,
with $Q=\mathfrak{a}_{\chi}$ and $s=-(it+v)$) and [27, Lemma 2] (when
$\mathfrak{a}_{\chi}=1$, with $Q=\mathfrak{a}_{\chi}$ and $s=-(it+v)$) for the
$\Gamma$-function. Both cases results in the bound

$|L(i(t+v),\chi)|\leq\sqrt{\frac{q_{\chi}}{2\pi}}|1+i(t+v)|^{\frac{1}{2}}|L(1+i(t+v),\chi)|.$

Since the growth of $\cosh(\pi v)$ in $v$ dominates the polynomial growth of
$\log|L(1+\xi+i(t+v),\chi)|$ in $v$, the above integral is at most

$\lim_{\xi\to
0^{+}}\int_{\mathbb{R}}\log|L(1+\xi+i(t+v),\chi)|\frac{dv}{\cosh(\pi
v)}+\frac{1}{4}\int_{\mathbb{R}}\log|1+i(t+v)|\frac{dv}{\cosh(\pi
v)}+\frac{1}{4}\log\frac{q_{\chi}}{2\pi}.$

By dominated convergence again, the limit equals

$\lim_{\xi\to
0^{+}}\textup{Re}\Big{(}\sum_{n=1}^{\infty}\frac{\Lambda(n)\chi(n)}{n^{1+\xi+it}\log
n}\int_{\mathbb{R}}n^{-iv}\frac{dv}{\cosh(\pi v)}\Big{)}\\\
=2\textup{Re}\Big{(}\sum_{n=1}^{\infty}\frac{\Lambda(n)\chi(n)}{(n^{\frac{3}{2}+it}+n^{\frac{1}{2}+it})\log
n}\Big{)}\leq 2\sum_{n=2}^{\infty}\frac{\Lambda(n)}{\log
n}\sum_{j=0}^{\infty}\frac{(-1)^{j}}{n^{\frac{3}{2}+j}}=2\sum_{j=0}^{\infty}(-1)^{j}\log\zeta(j+\tfrac{3}{2})\leq
1.4988.$

If $t\in\mathbb{R}$, then

$\frac{1}{4}\int_{\mathbb{R}}\log|1+i(t+v)|\frac{dv}{\cosh(\pi
v)}=\frac{1}{4}\log|1+it|+\frac{1}{8}\int_{\mathbb{R}}\log\Big{|}1+\frac{v^{2}+2|t|v}{t^{2}+1}\Big{|}\frac{dv}{\cosh(\pi
v)}.$

If $w\geq-1$, we see via Taylor expansions that

$\lim_{w_{0}\to w^{+}}\log|1+w_{0}|\leq w-\frac{w^{2}}{2}+\frac{w^{3}}{3}.$

For all $t,v\in\mathbb{R}$, we have that
$v^{2}+2|t|v=(v+|t|)^{2}-t^{2}\geq-t^{2}>-(t^{2}+1)$, so

$\frac{1}{8}\int_{\mathbb{R}}\log\Big{|}1+\frac{v^{2}+2|t|v}{t^{2}+1}\Big{|}\frac{dv}{\cosh(\pi
v)}\leq\frac{1}{8}\int_{\mathbb{R}}\sum_{j=1}^{3}\frac{(-1)^{j+1}}{j}\Big{(}\frac{v^{2}+2|t|v}{t^{2}+1}\Big{)}^{j}\frac{dv}{\cosh(\pi
v)}=\frac{79+210t^{2}-48t^{4}}{1536(t^{2}+1)^{3}}.$

Since

$\sup_{t\in\mathbb{R}}\exp\Big{(}\frac{1}{4}\log\frac{1}{2\pi}+1.4988+\frac{79+210t^{2}-48t^{4}}{1536(t^{2}+1)^{3}}\Big{)}\leq
2.97655,$

the result follows.

If $\chi=1$, then $L(s,\chi)=\zeta(s)$. In this case, the identity
$\zeta(\overline{s})=\overline{\zeta(s)}$ and [13, Theorem 1.1] yield

$|\zeta(\tfrac{1}{2}+it)|\leq\begin{cases}1.461&\mbox{if $0\leq|t|\leq 3$,}\\\
0.63|t|^{1/6}\log|t|&\mbox{if $|t|\geq 3$.}\end{cases}$

This finishes our proof in all cases. ∎

Following [11], we combine Proposition 2.10 with a version of Jensen’s formula
to improve our bounds for the sum over zeros in Lemma 2.9 in a more
restrictive range of $r$ than in Lemma 2.9.

######  Proposition 2.11.

Let $\chi\,(\mathrm{mod}\,\,q)$ be primitive, $T\geq 1$, $t\in\mathbb{R}$,
$|t|\leq T$, and $0<r<\frac{1}{2}$. If $\chi=1$, then assume that $3\times
10^{12}\leq|t|\leq T$. There holds

$\displaystyle\sum_{\begin{subarray}{c}L(\rho,\chi)=0\\\ |1+it-\rho|\leq
r\end{subarray}}\textup{Re}\Big{(}\frac{1}{1+r+it-\rho}\Big{)}$
$\displaystyle\leq\frac{\log(qT)+4.7098}{4+8r}+\frac{(\frac{4}{\pi}-1)\log(1+r^{-1})}{1+2r}+2.6908$
$\displaystyle+\frac{8r}{(1+2r)^{2}}(r(2\log(qT)-1)+4)+r^{-1}.$

###### Proof.

Let $s=1+r+it$. Choose $r_{1}\in(\max\\{\frac{1}{2},2r\\},\frac{1}{2}+r)$ so
that no zeros of $L(z,\chi)$ lie on the circle
$\\{w\in\mathbb{C}\colon|s-w|=r_{1}\\}$. By [11, Lemma 3.2] (with
$f(z)=(\frac{z-1}{z+1})^{\delta_{\chi}}L(z,\chi)$ and $a=s$), we have that if

$J:=\int_{0}^{2\pi}\frac{\cos\theta}{\pi
R}\log\Big{|}\Big{(}\frac{s+r_{1}e^{i\theta}-1}{s+r_{1}e^{i\theta}+1}\Big{)}^{\delta_{\chi}}L(s+r_{1}e^{i\theta},\chi)\Big{|}d\theta,$

then

$-\textup{Re}\Big{(}\frac{L^{\prime}}{L}(s,\chi)\Big{)}=\textup{Re}\Big{(}\frac{2\delta_{\chi}}{s^{2}-1}\Big{)}-\sum_{|s-\rho|<r_{1}}\textup{Re}\Big{(}\frac{1}{s-\rho}-\frac{s-\rho}{r_{1}^{2}}\Big{)}-J.$

Under our hypotheses, for $t$ and $T$ if $\chi=1$, we have the bound

$\delta_{\chi}\Big{|}\textup{Re}\Big{(}\frac{2}{s^{2}-1}\Big{)}\Big{|}=\delta_{\chi}\Big{|}\frac{r}{r^{2}+t^{2}}-\frac{2+r}{(2+r)^{2}+t^{2}}\Big{|}\leq\frac{\delta_{\chi}}{10^{24}}.$

Since $0<r<2r<r_{1}$ and $0<\beta<1$, it follows using nonnegativity that

$\displaystyle-\textup{Re}\Big{(}\frac{L^{\prime}}{L}(s,\chi)\Big{)}\leq\frac{\delta_{\chi}}{10^{25}}-\sum_{|1+it-\rho|\leq
r}\textup{Re}\Big{(}\frac{1}{s-\rho}\Big{)}+\sum_{|1+it-\rho|\leq
r}\textup{Re}\Big{(}\frac{s-\rho}{r_{1}^{2}}\Big{)}-J.$

If $|1+it-\rho|<r$, then $\textup{Re}(s-\rho)=1-\beta+r<2r$. Consequently, by
Lemma 2.9, we have that

$\sum_{|1+it-\rho|\leq
r}\textup{Re}\Big{(}\frac{s-\rho}{r_{1}^{2}}\Big{)}\leq\frac{2r}{r_{1}^{2}}n_{\chi}(r,1+it)\leq\frac{2r}{r_{1}^{2}}(r(2\log(qT)-1)+4).$

It remains to bound $|J|$ and choose $r_{1}$.

We will work with the decomposition

$J=\int_{0}^{\frac{\pi}{2}-r}+\int_{\frac{\pi}{2}-r}^{\frac{\pi}{2}}+\int_{\frac{\pi}{2}}^{\frac{3\pi}{2}}+\int_{\frac{3\pi}{2}}^{\frac{3\pi}{2}+r}+\int_{\frac{3\pi}{2}+r}^{2\pi}=J_{1}+J_{2}+J_{3}+J_{4}+J_{5}.$

We observe that if $\theta\in[0,\frac{\pi}{2}]\cup[\frac{3\pi}{2},2\pi]$, then
$\cos\theta\geq 0$ and

$\frac{\cos\theta}{\pi
r_{1}}\log\Big{|}\Big{(}\frac{s+r_{1}e^{i\theta}-1}{s+r_{1}e^{i\theta}+1}\Big{)}^{\delta_{\chi}}L(s+r_{1}e^{i\theta},\chi)\Big{|}\leq\frac{\cos\theta}{\pi
r_{1}}\log\zeta(1+r+r_{1}\cos\theta)\leq\frac{\cos\theta}{\pi
r_{1}}\log\frac{1+r+r_{1}\cos\theta}{r+r_{1}\cos\theta}.$

(We have used the fact that $\textup{Re}(s+r_{1}e^{i\theta})\geq 1+r$, so
$(\cos\theta)\log|\frac{s+r_{1}e^{i\theta}-1}{s+r_{1}e^{i\theta}+1}|\leq 0$ in
our current range of $\theta$.) Our hypotheses on $r$ and $r_{1}$ imply that
if $0<\theta<\frac{\pi}{2}$, then

$\frac{1}{\pi
r_{1}}\log\frac{1+r+r_{1}\cos\theta}{r+r_{1}\cos\theta}\leq\frac{2}{\pi}\log\frac{5}{2(r+r_{1}\cos\theta)}\leq\frac{2}{\pi}\log\Big{(}5\min\Big{\\{}\frac{1}{\cos\theta},\frac{1}{2r}\Big{\\}}\Big{)}.$

Therefore,

$J_{1}+J_{2}\leq\frac{2}{\pi}\Big{(}\int_{0}^{\frac{\pi}{2}-r}\Big{(}\log\frac{5}{\cos\theta}\Big{)}\cos\theta
d\theta+\Big{(}\log\frac{5}{2r}\Big{)}\int_{\frac{\pi}{2}-r}^{\frac{\pi}{2}}\cos\theta
d\theta\Big{)}\leq 1.3454.$

We similarly estimate $J_{4}+J_{5}$ using the symmetry of $\cos\theta$,
concluding that

$J_{1}+J_{2}+J_{4}+J_{5}\leq 2.6908.$

We estimate $J_{3}$ with a different approach. Let
$\theta\in[\frac{\pi}{2},\frac{3\pi}{2}]$. Since $2r<r_{1}<\frac{1}{2}+r$, we
have that $\frac{1}{2}\leq 1+r+r_{1}\cos\theta\leq 1+r$. Write
$\sigma^{\prime}=1+r+r_{1}\cos\theta$ and $y=t+r_{1}\sin\theta$ so that
$s+r_{1}e^{i\theta}=\sigma^{\prime}+iy$. Also, if

$F(\sigma^{\prime}+iy):=\Big{(}\frac{\sigma^{\prime}+iy-1}{\sigma^{\prime}+iy+1}\Big{)}^{\delta_{\chi}}L(\sigma^{\prime}+iy,\chi),$

then

$F(\sigma^{\prime}+iy)=\Big{(}\frac{s+r_{1}e^{i\theta}-1}{s+r_{1}e^{i\theta}+2}\Big{)}^{\delta_{\chi}}L(s+r_{1}e^{i\theta},\chi).$

We estimate $|F(\sigma^{\prime}+iy)|$ on the lines
$\sigma^{\prime}=\frac{1}{2}$ and $\sigma^{\prime}=1+r$ using Proposition 2.10
and the bound

$|F(1+r+iu)|\leq\zeta(1+r^{-1})\leq 1+r^{-1},\qquad u\in\mathbb{R},$

respectively. We estimate $F(\sigma^{\prime}+iy)$ in between these lines using
the Phragmén–Lindelöf principle; in particular,

$|F(\sigma^{\prime}+iy)|\leq(e^{\frac{1}{4}\log(q|1+it|)+1.0908})^{\frac{1+r-\sigma^{\prime}}{\frac{1}{2}+r}}(1+r^{-1})^{1-\frac{1+r-\sigma^{\prime}}{\frac{1}{2}+r}}.$

(We have used the fact that if $\textup{Re}(z)\geq\frac{1}{2}$, then
$|\frac{z-1}{z+1}|\leq 1$.) The ranges of $\theta$, $r$, and $r_{1}$ ensure
that

$\log\Big{|}(e^{\frac{1}{4}\log(q|1+it|)+1.0908})^{-\frac{2r_{1}\cos\theta}{1+2r}}(1+r^{-1})^{\frac{1+2r+2r_{1}\cos\theta}{1+2r}}\Big{|}\geq
0,$

so

$|J_{3}|\leq\frac{1}{\pi
r_{1}}\int_{\frac{\pi}{2}}^{\frac{3\pi}{2}}|\cos\theta|\cdot\log\Big{|}(e^{\frac{1}{4}\log(q|1+it|)+1.0908})^{-\frac{2r_{1}\cos\theta}{1+2r}}(1+r^{-1})^{\frac{1+2r+2r_{1}\cos\theta}{1+2r}}\Big{|}d\theta.$

Collecting our estimates for $J_{1},\ldots,J_{5}$, we find that

$\lim_{r_{1}\to(r+\frac{1}{2})^{-}}|J|\leq\frac{\log(qT)+4.709\,774}{4+8r}+\frac{(\frac{4}{\pi}-1)\log(1+r^{-1})}{1+2r}+2.6908.$

Since

$\frac{4.709\,774}{4+8r}+\frac{\delta_{\chi}}{10^{24}}\leq\frac{4.7098}{4+8r},\qquad\textup{Re}\Big{(}\frac{L^{\prime}}{L}(1+r+it,\chi)\Big{)}\leq-\frac{\zeta^{\prime}}{\zeta}(1+r)\leq\frac{1}{r}$

in our range of $r$, we arrive at the desired result. ∎

######  Corollary 2.12.

Let $Q\geq 3$, $T\geq 1$, and $\chi\,(\mathrm{mod}\,\,q)$ be a primitive
Dirichlet character with $q\leq Q$. Let $t\in\mathbb{R}$ and $|t|\leq T$. If
$\chi=1$, then assume that $3\times 10^{12}\leq|t|\leq T$. If $\chi\neq 1$,
then assume that $\max\\{Q,T\\}>10\,000$. If $n_{\chi}(r,s)$ is as in (2.6)
and

$\frac{1}{3\log(QT)}\leq r\leq\frac{1}{10},$

then $n_{\chi}(r,1+it)\leq r(1+10^{-7})^{-1}(\frac{2}{3}\log(QT)+13.04)+2$.

######  Remark.

The peculiar factor of $(1+10^{-7})^{-1}$ is for future convenience.

###### Proof.

By our hypotheses on $Q$ and $T$, the range of $\eta$ is nonempty. Let
$s=1+r+it$. If $\rho$ is a nontrivial zero of $L(s,\chi)$ and $|1+it-\rho|\leq
r$, then

$\frac{1}{2r}\leq\textup{Re}\Big{(}\frac{1}{1+r+it-\rho}\Big{)}.$

Therefore, by Proposition 2.11 (which assumes that $3\times 10^{12}\leq|t|\leq
T$ when $\chi=1$), we have that

$n_{\chi}(r,1+it)\leq
2r\Big{(}\frac{\log(qT)+4.7908}{4+8r}+\frac{(\frac{4}{\pi}-1)\log(1+r^{-1})}{1+2r}+2.6908\\\
+\frac{8r}{(1+2r)^{2}}(r(2\log(qT)-1)+4)+r^{-1}\Big{)}.$

If $\frac{1}{3\log(QT)}\leq r\leq\frac{1}{10}$, then

$\displaystyle
2\Big{(}\frac{\log(qT)+4.7908}{4+8r}+\frac{(\frac{4}{\pi}-1)\log(1+r^{-1})}{1+2r}+2.6908+\frac{8r}{(1+2r)^{2}}(r(2\log(qT)-1)+4)\Big{)}$
$\displaystyle\leq\frac{23}{36}\log(QT)+\frac{5}{3}\Big{(}\frac{4}{\pi}-1\Big{)}\log(1+3\log(QT))+11.7111.$

The result now follows since $\max\\{Q,T\\}>10\,000$. ∎

######  Remark.

Under the hypotheses of Corollary 2.12, our proof can be modified to produce
an upper bound of the form
$n_{\chi}(r,1+it)\leq\frac{1}{2}r\log(QT)+O(\log\log(QT))$ when
$\frac{1}{3\log(eQT)}\leq r\leq\frac{1}{30}$. But we cannot take full
advantage of this due to considerations involving application of the “pre-
sifted” large sieve inequality in Lemma 5.1 during our proof of Theorem 1.2 in
Section 5.

###  2.4. The Deuring–Heilbronn phenomenon

Here, we quantify the idea that if $\beta_{1}(Q)$ is close to $1$ in Theorem
2.6, then the zero-free region for all other Dirichlet $L$-functions
(including $\zeta(s)$) improves as a specific function of how close
$\beta_{1}(Q)$ is to $s=1$. This was first quantified by Linnik [19] for low-
lying zeros. Our proof relies on the following lower bound for power sums.

######  Lemma 2.13 (Theorem 2.2 of [17]).

Let $\varepsilon>0$. Let $(z_{n})_{n=1}^{\infty}$ (resp.
$(b_{n})_{n=1}^{\infty}$) be a sequence of complex (resp. nonnegative real)
numbers with $z_{1}\neq 0$ (resp. $b_{1}>0$). There exists an integer

$1\leq
j\leq(8+\varepsilon)\frac{1}{b_{1}}\sum_{n=1}^{\infty}\frac{b_{n}|z_{n}|}{|z_{1}|+|z_{n}|}$

such that

$\textup{Re}\Big{(}\sum_{n=1}^{\infty}b_{n}z_{n}^{j}\Big{)}\geq\frac{\varepsilon}{32+4\varepsilon}b_{1}|z_{1}|^{j}.$

######  Corollary 2.14.

Let $\varepsilon>0$. Let $(z_{n})_{n=1}^{\infty}$ be a sequence of complex
numbers with $z_{1}\neq 0$. Define
$\mathcal{M}:=\frac{1}{|z_{1}|}\sum_{n=1}^{\infty}|z_{n}|$. There exists an
integer $1\leq j\leq(8+\varepsilon)\mathcal{M}$ such that

$\textup{Re}\Big{(}\sum_{n=1}^{\infty}z_{n}^{j}\Big{)}\geq\frac{\varepsilon}{32+4\varepsilon}|z_{1}|^{j}.$

###### Proof.

Since $\frac{|z_{n}|}{|z_{1}|+|z_{n}|}\leq\frac{|z_{n}|}{|z_{1}|}$ when
$z_{1}\neq 0$, this follows from Lemma 2.13 with $b_{n}=1$ for all $n\geq 1$.
∎

We use Corollary 2.14 to prove what appears to be the first completely
explicit version of the Deuring–Heilbronn zero repulsion phenomenon.

######  Theorem 2.15.

Let $Q>400\,000$ and $T\geq 1$. Recall (1.1) and Theorem 2.6. Suppose that
$\beta_{1}(Q)\geq 1-\frac{\Cr{ZFR}}{\log Q}$, and let
$\chi_{1}\,(\mathrm{mod}\,\,q_{1})$ be the corresponding primitive quadratic
character with $q_{1}\in(400\,000,Q]$. If $\beta+i\gamma\neq\beta_{1}(Q)$ is a
nontrivial zero of $\mathscr{L}(s,Q)$ with $\beta>\frac{1}{2}$ and
$|\gamma|\leq T$, then

$\beta\leq
1-\frac{\log(\frac{1}{(1-\beta_{1}(Q))(670\,564.676+347\,029.502\log
Q+107\,906.278\log T)})}{104.645+54.156\log Q+16.84\log T}.$

######  Remark.

Our proof produces a stronger result than we have stated. For numerical
convenience, we assign specific values to the parameters $\varepsilon$ and
$\alpha$ occurring in our proof below.

######  Remark.

Jutila [16, Theorem 2] proved a numerically stronger result under the
assumption that $QT$ is sufficiently large (in terms of a parameter
$\varepsilon$ occurring in Jutila’s proof but not ours). To our knowledge,
Theorem 2.15 is the only explicit result of its type with no such hypotheses.

###### Proof.

The result is trivial when

$\beta_{1}(Q)\leq 1-\frac{1}{670\,564.676+347\,029.502\log Q+107\,906.278\log
T},$

so we assume otherwise. Let $1\leq q\leq Q$ be an integer, and let
$\chi\,(\mathrm{mod}\,\,q)$ be a primitive Dirichlet character. Let
$\beta+i\gamma\neq\beta_{1}(Q)$ be a nontrivial zero of $L(s,\chi)$ with
$\beta>\frac{1}{2}$ and $|\gamma|\leq T$, and let
$\psi\,(\mathrm{mod}\,\,q_{\psi})$ be the primitive character that induces
$\chi_{1}\chi$. Note that $q_{\psi}$ divides the least common multiple
$[q,q_{1}]$. Let $\omega$ denote a zero of the function

$D(s):=\zeta(s)L(s,\chi_{1})L(s+i\gamma,\chi)L(s+i\gamma,\chi_{1}\chi)=\zeta(s)L(s,\chi_{1})L(s+i\gamma,\chi)L(s+i\gamma,\psi)\prod_{p|[q,q_{1}]}\Big{(}1-\frac{\psi(p)}{p^{s+i\gamma}}\Big{)}.$

There exists numbers $a_{D},b_{D}\in\mathbb{C}$ such that

$(s-1)(s-1+i\gamma)^{\delta_{\chi}+\delta_{\psi}}D(s)=s^{\mathop{\mathrm{ord}}_{s=0}D(s)}e^{a_{D}+b_{D}s}\prod_{\begin{subarray}{c}\omega\neq
0\\\
D(\omega)=0\end{subarray}}\Big{(}1-\frac{s}{w}\Big{)}e^{\frac{s}{\omega}}.$

Let $\Lambda(n)$ be the von Mangoldt function, and define
$a(n):=(1+\chi_{1}(n))(1+\chi(n)n^{-i\gamma})\Lambda(n)$. By construction, we
have the uniform lower bound $\textup{Re}(a(n))\geq 0$. Also note that if
$\textup{Re}(s)>1$, then

$-\frac{D^{\prime}}{D}(s)=\sum_{n=1}^{\infty}\frac{a(n)\Lambda(n)}{n^{s}}.$

Let $\alpha\geq 1$. For an integer $j\geq 1$, the $(2j-1)$-th derivative at
$s=\alpha+1$ is

$\frac{1}{(2j-1)!}\sum_{n=1}^{\infty}\frac{a(n)\Lambda(n)(\log
n)^{2j-1}}{n^{\alpha+1}}=\frac{1}{\alpha^{2j}}+\frac{\delta_{\chi}+\mathbf{1}_{\psi=1}}{(\alpha+i\gamma)^{2j}}-\sum_{\omega}\frac{1}{(\alpha+1-\omega)^{2j}}.$

We extract the contribution from $\beta_{1}(Q)$, arriving at

$\frac{1}{(2j-1)!}\sum_{n=1}^{\infty}\frac{a(n)\Lambda(n)(\log
n)^{2j-1}}{n^{1+\alpha}}\\\
=\frac{1}{\alpha^{2j}}+\frac{\delta_{\chi}+\delta_{\psi}}{(\alpha+i\gamma)^{2j}}-\frac{1}{(\alpha+1-\beta_{1}(Q))^{2j}}-\frac{\delta_{\chi}+\delta_{\psi}}{(\alpha+1-\beta_{1}(Q)+i\gamma)^{2j}}-\sum_{n=1}^{\infty}z_{n}^{j},$

where $z_{n}=z_{n}(\gamma)$ runs through the (multi)sets

$\displaystyle\\{(1+\alpha-\omega)^{-2}\colon\zeta(\omega)=0\\},$
$\displaystyle\\{(1+\alpha-\omega)^{-2}\colon\omega\neq\beta_{1}(Q)\textup{
and }L(\omega,\chi_{1})=0\\},$
$\displaystyle\\{(1+\alpha+i\gamma-\omega)^{-2}\colon\omega\neq\beta_{1}(Q)\textup{
and }L(\omega,\chi)=0\\},$
$\displaystyle\\{(1+\alpha+i\gamma-\omega)^{-2}\colon\omega\neq\beta_{1}(Q)\textup{
and }L(\omega,\psi)=0\\},$ $\displaystyle\\{(1+\alpha+i\gamma+(h+2k)\pi i/\log
p)^{-2}\colon h\in\\{0,1\\},~{}k\in\mathbb{Z},~{}p\nmid q_{\psi}\\},$

arranged so that
$z_{1}=(\alpha+1+i\gamma-(\beta+i\gamma))^{-2}=(\alpha+1-\beta)^{-2}$. Since
$\textup{Re}(a(n))\geq 0$, it follows that

$\textup{Re}\Big{(}\frac{1}{(2j-1)!}\sum_{n=1}^{\infty}\frac{a(n)\Lambda(n)(\log
n)^{2j-1}}{n^{1+\alpha}}\Big{)}\geq 0.$

Since $\delta_{\chi}+\mathbf{1}_{\psi=1}\in\\{0,1\\}$, it follows that

$\displaystyle\textup{Re}\Big{(}\sum_{n=1}^{\infty}z_{n}^{j}\Big{)}$
$\displaystyle\leq\frac{1}{\alpha^{2j}}-\frac{1}{(\alpha+1-\beta_{1}(Q))^{2j}}+(\delta_{\chi}+\mathbf{1}_{\psi=1})\textup{Re}\Big{(}\frac{1}{(\alpha+i\gamma)^{2j}}-\frac{1}{(\alpha+1-\beta_{1}(Q)+i\gamma)^{2j}}\Big{)}$
$\displaystyle=\frac{1}{\alpha^{2j}}-\frac{1}{(\alpha+1-\beta_{1}(Q))^{2j}}+(\delta_{\chi}+\mathbf{1}_{\psi=1})\textup{Re}\Big{(}\frac{1}{(\alpha+i\gamma)^{2j}}\Big{(}1-\frac{1}{(1+\frac{1-\beta_{1}(Q)}{\alpha+i\gamma})^{2j}}\Big{)}\Big{)}$
$\displaystyle\leq\frac{1}{\alpha^{2j}}\Big{(}1-\frac{1}{(1+\frac{1-\beta_{1}(Q)}{\alpha})^{2j}}+\Big{|}1-\frac{1}{(1+\frac{1-\beta_{1}(Q)}{\alpha+i\gamma})^{2j}}\Big{|}\Big{)}.$

Since

$\Big{|}1-\frac{1}{(1+\frac{1-\beta_{1}(Q)}{\alpha+i\gamma})^{2j}}\Big{|}=\frac{2j}{|\alpha+i\gamma|}\Big{|}\int_{\beta_{1}(Q)}^{1}\frac{1}{(1+\frac{1-u}{\alpha+i\gamma})^{2j+1}}du\Big{|}\leq\frac{2j(1-\beta_{1}(Q))}{|\alpha+i\gamma|}\leq\frac{2j(1-\beta_{1}(Q))}{\alpha},$

regardless of whether or not $\gamma=0$, we find that

(2.7)
$\textup{Re}\Big{(}\sum_{n=1}^{\infty}z_{n}^{j}\Big{)}\leq\frac{4j(1-\beta_{1}(Q))}{\alpha^{2j+1}}.$

We will apply Corollary 2.14 to bound the left hand side of (2.7) from below.
Since $\beta>\frac{1}{2}$, we have that
$|z_{1}|\geq(\alpha+\frac{1}{2})^{-2}$, so
$\mathcal{M}:=|z_{1}|^{-1}\sum_{n=1}^{\infty}|z_{n}|$ is at most

$\Big{(}\alpha+\frac{1}{2}\Big{)}^{2}\Big{(}\sum_{\zeta(\omega)=0}\frac{1}{|\alpha+1-\omega|^{2}}+\sum_{\begin{subarray}{c}\omega\neq\beta_{1}(Q)\\\
L(\omega,\chi_{1})=0\end{subarray}}\frac{1}{|\alpha+1-\omega|^{2}}+\sum_{\begin{subarray}{c}\omega\neq\beta_{1}(Q)\\\
L(\omega,\chi)=0\end{subarray}}\frac{1}{|\alpha+1+i\gamma-\omega|^{2}}\\\
+\sum_{\begin{subarray}{c}\omega\neq\beta_{1}(Q)\\\
L(\omega,\psi)=0\end{subarray}}\frac{1}{|\alpha+1+i\gamma-\omega|^{2}}+\sum_{\begin{subarray}{c}p|[q,q_{1}]\\\
p\nmid
q_{\psi}\end{subarray}}\sum_{h=0}^{1}\sum_{k=-\infty}^{\infty}\frac{1}{|\alpha+1+i\gamma+\frac{(h+2k)\pi
i}{\log p}|^{2}}\Big{)}.$

We apply [34, Lemma 7.1] and its proof (with $K=\mathbb{Q}$) to bound the
contribution not arising from the nontrivial zeros; this contribution is at
most

$4\Big{(}\frac{1}{2(\alpha+1)}+\frac{1}{(\alpha+1)^{2}}\Big{)}+\Big{(}\frac{1}{2(\alpha+1)}+\frac{2}{(1+\alpha)^{2}\log
2}\Big{)}\log([q,q_{1}]).$

We apply [34, Equation (7–2)] (with $K=\mathbb{Q}$, which ensures that $\psi$
is quadratic) to bound the contribution arising from the nontrivial zeros. In
their proof (with $K=\mathbb{Q}$), they have $Q=\max_{\chi}q_{\chi}$ so that
$q,q_{1}\leq Q$ and $[q,q_{1}]\leq Q^{2}$. These simplifying estimates are
straightforward to reverse. We conclude (using the bound
$\delta_{\chi}+\mathbf{1}_{\chi_{1}\chi=1}\leq 1$) that

$\displaystyle\sum_{\zeta(\rho)=0}\frac{1}{|1+\alpha-\rho|^{2}}+\sum_{L(\rho,\chi_{1})=0}\frac{1}{|\alpha+1-\rho|}+\sum_{L(\rho,\chi)=0}\frac{1}{|\alpha+i+i\gamma-\rho|}+\sum_{L(\rho,\psi)=0}\frac{1}{|1+\alpha+i\gamma-\rho|}$
$\displaystyle\leq\frac{1}{\alpha}\textup{Re}\Big{(}\sum_{\zeta(\rho)=0}\frac{1}{1+\alpha-\rho}+\sum_{L(\rho,\chi_{1})=0}\frac{1}{\alpha+1-\rho}+\sum_{L(\rho,\chi)=0}\frac{1}{\alpha+i+i\gamma-\rho}+\sum_{L(\rho,\psi)=0}\frac{1}{1+\alpha+i\gamma-\rho}\Big{)}$
$\displaystyle\leq\frac{1}{\alpha}\Big{(}\frac{1}{2}\log(q_{1}q[q,q_{1}])+\frac{\log([q,q_{1}])}{2^{\alpha+1}-1}+\log(\alpha+2+|\gamma|)+\log(\alpha+2)+\frac{3}{1+\alpha}+\frac{1}{2^{\alpha+1}-1}-2\log\pi$
$\displaystyle+\frac{1}{\alpha}+\textup{Re}\Big{(}\frac{\delta_{\chi}+\mathbf{1}_{\chi_{1}\chi=1}}{\alpha+it}+\frac{\delta_{\chi}+\mathbf{1}_{\chi_{1}\chi=1}}{\alpha+1+it}\Big{)}\Big{)}$
$\displaystyle\leq\frac{1}{\alpha}\Big{(}\frac{1}{2}\log(q_{1}q[q,q_{1}])+\frac{\log([q,q_{1}])}{2^{\alpha+1}-1}+\log
T+2\log(\alpha+2)+\frac{6}{\alpha}+\log 2-2\log\pi\Big{)}$

When $\alpha\geq 12.8$, the above estimates (with $q,q_{1}\leq Q$ and
$[q,q_{1}]\leq Q^{2}$) together imply that

(2.8)
$\mathcal{M}\leq\frac{(\alpha+\frac{1}{2})^{2}}{\alpha}\Big{(}\frac{1}{2}\log(Q^{4}T^{2})+\Big{(}\frac{\alpha}{\alpha+1}+\frac{4\alpha}{(1+\alpha)^{2}\log
2}+\frac{2}{2^{\alpha+1}-1}\Big{)}\log
Q+2\log(\alpha+2)+\frac{1}{\alpha}\Big{)}.$

By Corollary 2.14 and a Taylor expansion argument, there exists
$j_{0}\in[1,(8+\varepsilon)\mathcal{M}]$ such that

$\textup{Re}\Big{(}\sum_{n=1}^{\infty}z_{n}^{j_{0}}\Big{)}\geq\frac{\varepsilon}{32+4\varepsilon}|z_{1}|^{j_{0}}=\frac{\varepsilon}{32+4\varepsilon}\cdot\frac{1}{(\alpha+1-\beta)^{2j_{0}}}\geq\frac{\varepsilon}{32+4\varepsilon}\alpha^{-2j_{0}}e^{-\frac{2j_{0}}{\alpha}(1-\beta)}.$

We deduce from this and (2.7) the bound
$\frac{\varepsilon}{32+4\varepsilon}e^{-2j_{0}(1-\beta)/\alpha}\leq
4j_{0}(1-\beta_{1}(Q))/\alpha$. Since
$j_{0}\in[1,(8+\varepsilon)\mathcal{M}]$, we rearrange to solve for $\beta$,
arriving at the bounds

$\beta\leq
1-\frac{\log(\frac{\varepsilon}{(1-\beta_{1}(Q))16(8+\varepsilon)^{2}\frac{\mathcal{M}}{\alpha}})}{2(8+\varepsilon)\frac{\mathcal{M}}{\alpha}},\qquad
1\leq\frac{16(8+\varepsilon)^{2}}{\varepsilon}(1-\beta_{1}(Q))\frac{\mathcal{M}}{\alpha}e^{2(8+\varepsilon)\frac{\mathcal{M}}{\alpha}(1-\beta)}.$

These two bounds imply the desired results once we use (2.8) with
$\varepsilon=\frac{1}{40}$ and $\alpha=23.22$ (and for the second part, we use
$T=Q>400\,000$). ∎

##  3\. Notation, conventions, and the key optimization

Throughout our proof of Theorem 1.2, we use the following notation and
conventions that are crucial to our proofs:

  1. (i)

$T\geq 1$, $Q\geq 1$, $1\leq q\leq Q$

  2. (ii)

$\mathcal{L}=\log(QT)$,

  3. (iii)

$\tau\in\mathbb{R}$ with $|\tau|\leq T$,

  4. (iv)

$\alpha=7.931\,643\,766\,252\,22\ldots$ (decimal expansion of a number whose
origin is explained below),

  5. (v)

$A=3.907\,668\,327\,378\,39\ldots$ (decimal expansion of a number whose origin
is explained below),

  6. (vi)

$R=\sqrt{A^{2}+1}$,

  7. (vii)

$\chi\,(\mathrm{mod}\,\,q)$ a primitive Dirichlet character,

  8. (viii)

$\frac{1}{3A\mathcal{L}}\leq\eta\leq\frac{1}{10A}$,

  9. (ix)

$s_{0}=1+\eta+i\tau$,

  10. (x)

$\max\\{Q,T\\}>10^{4}$,

  11. (xi)

$\delta_{\chi}$ the indicator function of the trivial primitive character,

  12. (xii)

if $\delta_{\chi}=1$, then $3\times 10^{12}\leq|\tau|\leq T$,

  13. (xiii)

$\mathbf{1}_{I}(t)$ the indicator function of a subinterval
$I\subseteq\mathbb{R}$,

  14. (xiv)

$\mathcal{N}_{\eta}=A\eta(1+10^{-7})^{-1}(\frac{2}{3}\mathcal{L}+13.04)+9$,

  15. (xv)

$M_{\eta}=(\alpha-1)\mathcal{N}_{\eta}$, which is at least $63.9$

  16. (xvi)

$k\in[M_{\eta},M_{\eta}+\mathcal{N}_{\eta}-1]\cap\mathbb{Z}\subseteq[M_{\eta},\frac{\alpha}{\alpha-1}M_{\eta}]\cap\mathbb{Z}$,
which implies that $k\geq 64$,

  17. (xvii)

$V=2(4e\alpha)^{1/(\alpha-1)}+0.38=4.184\,416\,849\ldots$,

  18. (xviii)

$A_{0}=1/(eV)=0.087\,916\,537\,56\ldots$,

  19. (xix)

$A_{1}>2$ satisfies $V^{-1}=A_{1}e^{1-\frac{A_{1}(\alpha-1)}{2\alpha}}$, so
$A_{1}=11.065\,510\,190\ldots$,

  20. (xx)

$N_{\eta}=\exp(A_{0}M_{\eta}/\eta)$,

  21. (xxi)

$N_{\eta}^{*}=\exp(A_{1}M_{\eta}/\eta)$,

  22. (xxii)

$\xi=1+10^{-7}$.

When we recall one of these definitions or conventions, we will identify them
as (i), (ii), (iii), etc.

The constants that arise in Corollary 2.12 (see (xiv)) and the numbers
$\alpha$ in (iv) and $A$ in (v) are central to these definitions and
conventions. Our values of $\alpha$ and $A$ arise as the values that

(3.1) minimize $(4e\alpha 2^{\alpha-1})^{A}$ with the constraints $\alpha>1$,
$A>1$, and
$4e\alpha\Big{(}\frac{2}{\sqrt{A^{2}+1}}\Big{)}^{\alpha-1}=\frac{2}{3}$.

This optimization problem serves as the basis for the zero detection process
in Section 4 (see (LABEL:eqn:lowerbound1) below). We could repeat our analysis
with $\frac{2}{3}$ replaced by any fixed constant $0<\delta<1$, and the
ensuing values of $\alpha$ and $A$ would depend on $\delta$. As $\delta\to
1^{-}$, these new values of $\alpha$ and $A$ would lead to a versions of
Theorem 1.2 with worse implied constants and better exponents. As $\delta\to
0^{+}$, the implied constants are better and the exponents are worse. One
cannot take $\delta$ to be too small because of considerations arising from
our eventual use of the “pre-sifted” large sieve inequality in Lemma 5.1. Our
choice of $\delta=\frac{2}{3}$ is a convenient middle ground between these
competing interests that does not affect the quality of the power savings in
Theorem 1.1.

The optimal choice of $\delta$ depends on the specific setting to which
Theorem 1.2 is applied. For any fixed value $0<\delta<1$, our method will
produce an explicit zero density estimate, so the interested reader can follow
our ideas with the values of $\alpha$ and $A$ arising from a different value
of $\delta$ and arrive at explicit versions of (1.3) and (1.5) with different
constants. We do not carry the dependence on $\delta$ throughout our proofs
for clarity of exposition.

There is some flexibility in our choices. We give two examples. For reasons
that will naturally arise in our proof, it is important that $V$ in (xvii) be
greater than $2(4e\alpha)^{1/(\alpha-1)}$. A large value of $V$ inflates the
implied constants in Theorem 1.2, but a small value of $V$ that is too close
to $2(4e\alpha)^{1/(\alpha-1)}$ will force us to needlessly inflate
$\mathcal{N}_{\eta}$. Our value of $V$ is convenient; there is no canonical
choice. All that matters for $\xi$ is that it is greater than 1, and the
quality of the exponent (resp. the implied constant) in Theorem 1.2 improves
(resp. worsens) as $\xi\to 1^{+}$.

##  4\. The main zero detection result

The reader is strongly encouraged to frequently consult the notation and
conventions in Section 3. We proceed to detect the zeros of $L(s,\chi)$ that
satisfy the following hypothesis.

######  Hypothesis 4.1.

Recall the notation and conventions in Section 3. There exists a nontrivial
zero $\rho_{0}$ of $L(s,\chi)$ such that $|1+i\tau-\rho_{0}|\leq\eta$.

We believe that 4.1 is false. Indeed, it contradicts GRH. Our main result en
route to Theorem 1.2 is the following zero detection result.

######  Theorem 4.2.

Recall the notation and conventions in Section 3. If 4.1 is true, then

$1\leq\frac{\eta^{3}}{200}M_{\eta}e^{2.672\,32M_{\eta}}\int_{N_{\eta}}^{N_{\eta}^{*}}\Big{|}\sum_{N_{\eta}<p\leq
u}\frac{\chi(p)\log p}{p^{1+i\tau}}\Big{|}^{2}\frac{du}{u}.$

######  Remark.

We remind the reader that if $\chi=1$, then one must have $3\times
10^{12}\leq|\tau|\leq T$ per (xii); otherwise, 4.1 is false by Lemma 2.1. If
$\chi\neq 1$, then one must have $\max\\{q,T\\}>10^{4}$ per (x); otherwise,
4.1 is false by Lemma 2.2.

Our proof relies on another lower bound for power sums, different from Lemma
2.13. The result that we cite, due to Kolesnik and Strauss, descends from
original arguments of Sós and Turán [33].

######  Lemma 4.3 ([18]).

Let $M\geq 0$ and $N\geq 1$ be integers, and let
$z_{1},\ldots,z_{N}\in\mathbb{Z}$. There exists an integer $k\in[M+1,M+N]$
such that

$\ |z_{1}^{k}+\cdots+z_{N}^{k}|\geq
1.007\Big{(}\frac{1}{4e(1+\frac{M}{N})}\Big{)}^{N}|z_{1}|^{k}.$

######  Remark.

The constant $4e$ is essentially optimal; see [21].

###  4.1. A conditional lower bound on high derivatives

Assume 4.1. By (2.2), if $\textup{Re}(s)>1$, then for $k$ satisfying (xvi), we
have that

$\frac{(-1)^{k+1}}{k!}\Big{(}\frac{L^{\prime}}{L}(s,\chi)\Big{)}^{(k)}=\delta_{\chi}\Big{(}\frac{1}{s^{k+1}}+\frac{1}{(s-1)^{k+1}}\Big{)}-\sum_{\rho}\frac{1}{(s-\rho)^{k+1}}-\sum_{n=0}^{\infty}\frac{1}{(s+2(n+\delta_{\chi})+\mathfrak{a}_{\chi})^{k+1}}.$

We evaluate the above expression at $s=s_{0}$, with $s_{0}$ as in (ix). Some
casework shows that

$\Big{|}\frac{\delta_{\chi}}{s_{0}^{k+1}}-\sum_{n=0}^{\infty}\frac{1}{(s_{0}+2(n+\delta_{\chi})+\mathfrak{a}_{\chi})^{k+1}}\Big{|}\leq(1-2^{-k-1})\zeta(k+1)\leq
1+k^{-2}.$

By (xii), we have that

$\delta_{\chi}\Big{|}\frac{1}{(s_{0}-1)^{k+1}}\Big{|}\leq\frac{\delta_{\chi}}{10^{12(k+1)}}.$

If $|1+i\tau-\rho|>A\eta$, then $|s_{0}-\rho|\geq R\eta$. Therefore, by an
intermediate calculation in the proof of Lemma 2.9, we have that

$\displaystyle\sum_{|1+i\tau-\rho|>A\eta}\frac{1}{|s_{0}-\rho|^{k+1}}$
$\displaystyle\leq\frac{1}{(R\eta)^{k-1}}\sum_{\rho}\frac{1}{|s_{0}-\rho|^{2}}$
$\displaystyle\leq\frac{1}{(R\eta)^{k-1}\eta}\sum_{\rho}\frac{1+\eta-\beta}{|s_{0}-\rho|^{2}}=\frac{R}{(R\eta)^{k}}\sum_{\rho}\textup{Re}\Big{(}\frac{1}{s_{0}-\rho}\Big{)}\leq\frac{(\eta\mathcal{L}+2)R^{2}}{2(R\eta)^{k+1}}.$

Combining the above results in this subsection and applying the lower bound on
$k$ in (xvi), we arrive at the bound

(4.1)
$\displaystyle\frac{\eta^{k+1}}{k!}\Big{|}\Big{(}\frac{L^{\prime}}{L}\Big{)}^{(k)}(s_{0},\chi)\Big{|}$
$\displaystyle\geq\Big{|}\sum_{|1+i\tau-\rho|\leq
A\eta}\frac{\eta^{k+1}}{(s_{0}-\rho)^{k+1}}\Big{|}-\frac{1}{R^{k+1}}\Big{(}\frac{\eta\mathcal{L}+2}{2}R^{2}+(1+k^{-2}+10^{-12(k+1)})(R\eta)^{k+1}\Big{)}$
$\displaystyle\geq\Big{|}\sum_{|1+i\tau-\rho|\leq
A\eta}\frac{\eta^{k+1}}{(s_{0}-\rho)^{k+1}}\Big{|}-\frac{1}{R^{k+1}}\Big{(}\frac{\eta\mathcal{L}+2}{2}R^{2}+1.001(R\eta)^{k+1}\Big{)}.$

By 4.1, the sum over zeros in (4.1) is nonempty, which makes Lemma 4.3
applicable. In the notation of Lemma 4.3, we have that

$N=n_{\chi}(A\eta,1+i\tau)\geq
1,\qquad\\{z_{1},\ldots,z_{N}\\}=\\{(s_{0}-\rho)^{-1}\colon|1+i\tau-\rho|\leq
A\eta\\},\qquad z_{1}=\rho_{0}.$

Note that $N\leq\mathcal{N}_{\eta}$ (see (xiv)) per Corollary 2.12. With
$M_{\eta}$ as in (xv), we choose

$M=M_{\eta}.$

Per 4.1, we have that $|s_{0}-\rho_{0}|=|(1+i\tau-\rho_{0})+\eta|\leq 2\eta$.
It follows that there exists $k\in[M,M+N-1]$, necessarily greater than or
equal to $64$ per (xvi), such that

$\Big{|}\sum_{|1+i\tau-\rho|\leq
A\eta}\frac{\eta^{k+1}}{(s_{0}-\rho)^{k+1}}\Big{|}\geq
1.007\Big{(}\frac{1}{4e(1+\frac{M}{N})}\Big{)}^{N}\frac{\eta^{k+1}}{|s_{0}-\rho_{0}|^{k+1}}\geq\frac{1.007}{2^{k+1}}\Big{(}\frac{1}{4e(1+\frac{M}{N})}\Big{)}^{N}.$

This lower bound and (4.1) imply that

(4.2)
$\frac{\eta^{k+1}}{k!}\Big{|}\Big{(}\frac{L^{\prime}}{L}\Big{)}^{(k)}(s_{0},\chi)\Big{|}\geq\frac{1.007}{2^{k+1}}\Big{(}\frac{1}{4e(1+\frac{M}{N})}\Big{)}^{N}-\frac{1}{R^{k+1}}\Big{(}\frac{\eta\mathcal{L}+2}{2}R^{2}+1.001(R\eta)^{k+1}\Big{)}.$

Using the bound $k+1\geq M_{\eta}=(\alpha-1)\mathcal{N}_{\eta}$ as well as
(vi), (x), (xiv), (xv), and (xvi), we find that (4.2) is at least

(4.3) $\displaystyle
1.007\Big{(}\frac{1}{4e\alpha}\Big{)}^{\mathcal{N}_{\eta}}\frac{1}{2^{k+1}}-\frac{1}{R^{k+1}}\Big{(}\frac{\eta\mathcal{L}+2}{2}R^{2}+1.001(R\eta)^{k+1}\Big{)}$
$\displaystyle\geq
1.007\Big{(}\frac{1}{4e\alpha}\Big{)}^{\mathcal{N}_{\eta}}\frac{1}{2^{k+1}}\Big{(}1-(4e\alpha)^{\mathcal{N}_{\eta}}\frac{2^{k+1}}{R^{k+1}}3.123\mathcal{N}_{\eta}\Big{)}$
$\displaystyle\geq 1.007\Big{(}\frac{1}{(4e\alpha
2^{\alpha-1})^{A}}\Big{)}^{\frac{\mathcal{N}_{\eta}}{A}}\Big{(}1-\Big{(}4e\alpha\Big{(}\frac{2}{\sqrt{A^{2}+1}}\Big{)}^{\alpha-1}\Big{)}^{\mathcal{N}_{\eta}}3.123\mathcal{N}_{\eta}\Big{)}.$

It is at this point that the minimization problem (3.1) arises, and our
choices of $\alpha$ in (iv) and $A$ in (v) are justified. (Notice that the
coefficient of $\eta\mathcal{L}$ in $\mathcal{N}_{\eta}/A$ is independent of
$A$.) Now, (LABEL:eqn:lowerbound1) equals

(4.4) $\displaystyle 1.007\Big{(}\frac{1}{(4e\alpha
2^{\alpha-1})^{\frac{1}{\alpha-1}}}\Big{)}^{M_{\eta}}\Big{(}1-\Big{(}\frac{2}{3}\Big{)}^{\mathcal{N}_{\eta}}3.123\mathcal{N}_{\eta}\Big{)}$
$\displaystyle=1.007\Big{(}\frac{1}{3.804\,416\,849\,672\ldots}\Big{)}^{M_{\eta}}\Big{(}1-\Big{(}\frac{2}{3}\Big{)}^{\mathcal{N}_{\eta}}3.123\mathcal{N}_{\eta}\Big{)}.$

Since $\mathcal{N}_{\eta}\geq 9$ per (xiv), it follows that

(4.5)
$\frac{\eta^{k+1}}{k!}\Big{|}\Big{(}\frac{L^{\prime}}{L}\Big{)}^{(k)}(s_{0},\chi)\Big{|}\geq
0.27\Big{(}\frac{1}{3.804\,416\,849\,673}\Big{)}^{M_{\eta}}.$

###  4.2. Unconditional upper bounds for high derivatives

Now, we turn to an upper bound for

(4.6)
$\frac{\eta^{k+1}}{k!}\Big{|}\Big{(}\frac{L^{\prime}}{L}\Big{)}^{(k)}(s_{0},\chi)\Big{|}=\eta\Big{|}\sum_{n=1}^{\infty}\frac{\Lambda(n)\chi(n)}{n^{1+i\tau}}j_{k}(\eta\log
n)\Big{|},\qquad j_{k}(u)=e^{-u}\frac{u^{k}}{k!}.$

Per (xvi), we have that
$k\in[(\alpha-1)\mathcal{N}_{\eta},\alpha\mathcal{N}_{\eta}]\subseteq[M_{\eta},\frac{\alpha}{\alpha-1}M_{\eta}]$
and $k\geq 64$. Recall the definitions of $V$ (xvii), $A_{0}$ (xviii), $A_{1}$
(xix), $N_{\eta}$ (xx), and $N_{\eta}^{*}$ (xxi). We decompose the right hand
side of (4.6) into a sum over $n\notin(N_{\eta},N_{\eta}^{*}]$, a sum over
composite $n\in(N_{\eta},N_{\eta}^{*}]$, and a (dominant) sum over prime
$n\in(N_{\eta},N_{\eta}^{*}]$.

If $n\leq N_{\eta}$, then $\eta\log n\leq A_{0}M_{\eta}$. Since
$k\in[M_{\eta},\frac{\alpha}{\alpha-1}M_{\eta}]$ and $k!\geq(k/e)^{k}$, it
follows that

$j_{k}(\eta\log n)=n^{-\eta}\frac{(\eta\log n)^{k}}{k!}\leq
n^{-\eta}\Big{(}\frac{A_{0}eM_{\eta}}{k}\Big{)}^{k}\leq n^{-\eta}V^{-k}.$

If $n\geq N_{\eta}^{*}$, then $\eta\log n\geq A_{1}M_{\eta}$. Since
$e^{-u/2}u^{k}/k!$ is decreasing for $u>2k$ and $A_{1}>2$, we find that if
$n\geq N_{\eta}^{*}$, then

$\displaystyle j_{k}(\eta\log
n)=n^{-\frac{\eta}{2}}\frac{e^{-\frac{1}{2}\eta\log n}(\eta\log
n)^{k}}{k!}\leq
n^{-\frac{\eta}{2}}\frac{e^{-\frac{1}{2}A_{1}M_{\eta}}(A_{1}M_{\eta})^{k}}{k!}$
$\displaystyle\leq
n^{-\frac{\eta}{2}}e^{-\frac{1}{2}A_{1}M_{\eta}}(A_{1}M_{\eta})^{k}\Big{(}\frac{e}{k}\Big{)}^{k}$
$\displaystyle=n^{-\frac{\eta}{2}}(A_{1}e^{1-\frac{1}{2}A_{1}\frac{M_{\eta}}{k}})^{k}\Big{(}\frac{M_{\eta}}{k}\Big{)}^{k}$
$\displaystyle\leq
n^{-\frac{\eta}{2}}(A_{1}e^{1-\frac{\alpha-1}{2\alpha}A_{1}})^{k}=n^{-\frac{\eta}{2}}V^{-k}.$

Therefore, we have the bound

$\eta\Big{|}\sum_{n\notin(N_{\eta},N_{\eta}^{*}]}\frac{\Lambda(n)\chi(n)}{n^{1+i\tau}}j_{k}(\eta\log
n)\Big{|}\leq\frac{\eta}{V^{k}}\Big{(}\sum_{n=1}^{\infty}\frac{\Lambda(n)}{n^{1+\eta}}+\sum_{n=1}^{\infty}\frac{\Lambda(n)}{n^{1+\frac{\eta}{2}}}\Big{)}\leq\frac{\eta}{V^{k}}\Big{(}\frac{1}{\eta}+\frac{2}{\eta}\Big{)}=\frac{3}{V^{k}}.$

Now, let $n\in(N_{\eta},N_{\eta}^{*}]$. Since $(\log u)^{k}\leq k!u$ for
$k,u\geq 1$ and $0<\eta\leq\frac{1}{10A}$ per (viii), we have that

$j_{k}(\eta\log n)=n^{-\eta}\frac{(\eta\log
n)^{k}}{k!}=n^{-\eta}(2\eta)^{k}\frac{(\log
n^{\frac{1}{2}})^{k}}{k!}\leq(2\eta)^{k}n^{\frac{1}{2}-\eta}\leq\frac{n^{\frac{1}{2}-\eta}}{(3V)^{k}}.$

Thus,

$\displaystyle\eta\Big{|}\sum_{\begin{subarray}{c}n\in(N_{\eta},N_{\eta}^{*}]\\\
\textup{$n$
composite}\end{subarray}}\frac{\Lambda(n)\chi(n)}{n^{1+i\tau}}j_{k}(\eta\log
n)\Big{|}$
$\displaystyle\leq\frac{\eta}{(3V)^{k}}\sum_{\begin{subarray}{c}n\in(N_{\eta},N_{\eta}^{*}]\\\
\textup{$n$ composite}\end{subarray}}\frac{\Lambda(n)}{n^{\frac{1}{2}+\eta}}$
$\displaystyle=\frac{\eta}{(3V)^{k}}\sum_{p}\sum_{j=2}^{\infty}\frac{\log
p}{p^{(\frac{1}{2}+\eta)j}}$
$\displaystyle\leq(2+\sqrt{2})\frac{\eta}{(3V)^{k}}\sum_{p}\frac{\log
p}{p^{1+2\eta}}$
$\displaystyle\leq-(2+\sqrt{2})\frac{\eta}{(3V)^{k}}\frac{\zeta^{\prime}}{\zeta}(1+2\eta)\leq(2+\sqrt{2})\frac{\eta}{(3V)^{k}}\frac{1}{2\eta}=\frac{1+2^{-\frac{1}{2}}}{(3V)^{k}}.$

Finally, we use partial summation to consider the sum

$\eta\Big{|}\sum_{p\in(N_{\eta},N_{\eta}^{*}]}\frac{\Lambda(p)\chi(p)}{p^{1+i\tau}}j_{k}(\eta\log
p)\Big{|}=\eta\Big{|}j_{k}(\eta\log
N_{\eta}^{*})\sum_{p\in(N_{\eta},N_{\eta}^{*}]}\frac{\chi(p)\log
p}{p^{1+i\tau}}\\\
-\int_{N_{\eta}}^{N_{\eta}^{*}}\sum_{p\in(N_{\eta},u]}\frac{\chi(p)\log
p}{p^{1+i\tau}}\Big{(}\frac{d}{du}j_{k}(\eta\log u)\Big{)}du\Big{|}.$

Our earlier bound for $j_{k}(\eta\log n)$ when $n\geq N_{\eta}^{*}$ implies
that

$\eta\Big{|}j_{k}(\eta\log
N_{\eta}^{*})\sum_{p\in(N_{\eta},N_{\eta}^{*}]}\frac{\chi(p)\log
p}{p^{1+i\tau}}\Big{|}\leq\frac{\eta(N_{\eta}^{*})^{-\frac{\eta}{2}}}{V^{k}}\sum_{p\leq
N_{\eta}^{*}}\frac{\log p}{p}\leq\frac{\eta}{V^{k}}\sum_{p\leq
N_{\eta}^{*}}\frac{\log p}{p^{1+\frac{\eta}{2}}}\leq\eta\frac{2e}{\eta
V^{k}}\leq\frac{2e}{V^{k}}.$

Since $k\geq 64$, it follows that

$\Big{|}\frac{d}{du}j_{k}(\eta\log u)\Big{|}=\frac{\eta}{u}|(j_{k-1}(\eta\log
u)-j_{k}(\eta\log u))|\leq\frac{\eta}{u}\sup_{\begin{subarray}{c}u\geq 1\\\
k\geq 64\\\ \eta>0\end{subarray}}\frac{u^{-\eta}(\eta\log u)^{k-1}|k-\eta\log
u|}{k!}\leq\frac{\eta}{200u}.$

We combine the results in this subsection to conclude that

(4.7)
$\frac{\eta^{k+1}}{k!}\Big{|}\Big{(}\frac{L^{\prime}}{L}\Big{)}^{(k)}(s_{0},\chi)\Big{|}\leq\frac{\eta^{2}}{200}\int_{N_{\eta}}^{N_{\eta}^{*}}\Big{|}\sum_{N_{\eta}<p\leq
u}\frac{\chi(p)\log p}{p^{1+i\tau}}\Big{|}\frac{du}{u}+\frac{8.437}{V^{k}}.$

Therefore,

(4.8)
$\frac{\eta^{k+1}}{k!}\Big{|}\Big{(}\frac{L^{\prime}}{L}\Big{)}^{(k)}(s_{0},\chi)\Big{|}\leq\frac{\eta^{2}}{200}\int_{N_{\eta}}^{N_{\eta}^{*}}\Big{|}\sum_{N_{\eta}<p\leq
u}\frac{\chi(p)\log p}{p^{1+i\tau}}\Big{|}\frac{du}{u}+\frac{8.437}{V^{k}}.$

###  4.3. Proof of Theorem 4.2

Under 4.1, the lower bound (4.5) holds. Unconditionally, the upper bound (4.8)
holds. Therefore, under 4.1, the two bounds imply that

$0.27\Big{(}\frac{1}{3.804\,416\,849\,673}\Big{)}^{M_{\eta}}\leq\frac{\eta^{2}}{200}\int_{N_{\eta}}^{N_{\eta}^{*}}\Big{|}\sum_{N_{\eta}<p\leq
u}\frac{\chi(p)\log p}{p^{1+i\tau}}\Big{|}\frac{du}{u}+\frac{8.437}{V^{k}}.$

It follows from our lower bounds for $k$ and $M_{\eta}$ in (xv) and (xvi) as
well as our value of $V$ in (xvii) that

$\frac{1}{4}e^{-1.336\,162\,721M_{\eta}}\leq\frac{\eta^{2}}{200}\int_{N_{\eta}}^{N_{\eta}^{*}}\Big{|}\sum_{N_{\eta}<p\leq
u}\frac{\chi(p)\log p}{p^{1+i\tau}}\Big{|}\frac{du}{u}.$

We square both sides and apply the Cauchy–Schwarz inequality to obtain

$\displaystyle\frac{1}{16}e^{-2.672\,32M_{\eta}}$
$\displaystyle\leq\Big{(}\frac{\eta^{2}}{200}\int_{N_{\eta}}^{N_{\eta}^{*}}\Big{|}\sum_{N_{\eta}<p\leq
u}\frac{\chi(p)\log p}{p^{1+i\tau}}\Big{|}\frac{du}{u}\Big{)}^{2}$
$\displaystyle\leq\frac{\eta^{4}}{40\,000}\Big{(}\int_{N_{\eta}}^{N_{\eta}^{*}}\frac{du}{u}\Big{)}\int_{N_{\eta}}^{N_{\eta}^{*}}\Big{|}\sum_{N_{\eta}<p\leq
u}\frac{\chi(p)\log p}{p^{1+i\tau}}\Big{|}^{2}\frac{du}{u}$
$\displaystyle\leq\frac{11}{40\,000}\eta^{3}M_{\eta}\int_{N_{\eta}}^{N_{\eta}^{*}}\Big{|}\sum_{N_{\eta}<p\leq
u}\frac{\chi(p)\log p}{p^{1+i\tau}}\Big{|}^{2}\frac{du}{u}.$

Theorem 4.2 follows once we multiply both sides by $16e^{2.672\,32M_{\eta}}$.

###  4.4. From detecting zeros to counting zeros

We apply Theorem 4.2 to count zeros of $L(s,\chi)$ that satisfy
$|1+i\tau-\rho|\leq\eta$. In order to account for zeros that lie on the
boundary, we increase the radius by a factor of $\xi$ (see (xxii)). Therefore,
in all that follows, every instance of $\eta$ from before must be increased by
a factor of $\xi$.

Let $\rho=\beta+i\gamma$ be a nontrivial zero of $L(s,\chi)$. For $w>1$,
define

$\Upphi_{\rho,w}(\tau)=\Upphi_{\rho,w,\chi}(\tau)=\begin{cases}1&\mbox{if
$|1+i\tau-\rho|\leq w$,}\\\ 0&\mbox{otherwise.}\end{cases}$

Regardless of whether $\rho_{0}$ satisfies 4.1, it follows that

$\Upphi_{\rho_{0},\xi\eta}(\tau)\leq\Big{(}\frac{1}{200}(\xi\eta)^{3}M_{\xi\eta}e^{2.672\,32M_{\xi\eta}}\int_{N_{\xi\eta}}^{N_{\xi\eta}^{*}}\Big{|}\sum_{N_{\xi\eta}<p\leq
u}\frac{\chi(p)\log
p}{p^{1+i\tau}}\Big{|}^{2}\frac{du}{u}\Big{)}\Upphi_{\rho_{0},\xi\eta}(\tau).$

We integrate both sides over $\tau\in[-T,T]$ and obtain

$\int_{-T}^{T}\Upphi_{\rho_{0},\xi\eta}(\tau)d\tau\leq\frac{1}{200}(\xi\eta)^{3}M_{\xi\eta}e^{2.672\,32M_{\xi\eta}}\int_{-T}^{T}\Big{(}\int_{N_{\xi\eta}}^{N_{\xi\eta}^{*}}\Big{|}\sum_{N_{\xi\eta}<p\leq
u}\frac{\chi(p)\log
p}{p^{1+i\tau}}\Big{|}^{2}\frac{du}{u}\Big{)}\Upphi_{\rho_{0},\xi\eta}(\tau)d\tau.$

Because of Lemma 2.1, this maneuver is acceptable even with (xii). If
$1-\eta\leq\textup{Re}(\rho_{0})<1$ and $|\textup{Im}(\rho_{0})|\leq T$, then
under 4.1, we have that

$\int_{-T}^{T}\Upphi_{\rho_{0},\xi\eta}(\tau)d\tau\geq\eta\sqrt{\xi^{2}-1}.$

Consequently, if $1-\eta\leq\textup{Re}(\rho_{0})<1$ and
$|\textup{Im}(\rho_{0})|\leq T$, then by Theorem 4.2,

(4.9)
$1\leq\frac{\xi^{3}}{200\sqrt{\xi^{2}-1}}\eta^{2}M_{\xi\eta}e^{2.672\,32M_{\xi\eta}}\int_{-T}^{T}\Big{(}\int_{N_{\xi\eta}}^{N_{\xi\eta}^{*}}\Big{|}\sum_{N_{\xi\eta}<p\leq
u}\frac{\chi(p)\log
p}{p^{1+i\tau}}\Big{|}^{2}\frac{du}{u}\Big{)}\Upphi_{\rho_{0},\xi\eta}(\tau)d\tau.$

Corollary 2.12, (xiv), and (xv) imply that

$\sum_{\begin{subarray}{c}\rho\\\
L(\rho,\chi)=0\end{subarray}}\Upphi_{\rho}(\tau)=n_{\chi}(\xi\eta,1+i\tau)\leq\frac{\xi}{\alpha-1}M_{\xi\eta}.$

Therefore, upon summing (4.9) over all nontrivial zeros $\rho=\beta+i\gamma$
of $L(s,\chi)$ with $1-\eta\leq\beta\leq 1$ and $|\gamma|\leq T$, we find that
if

$N_{\chi}(\sigma,T):=\\#\\{\rho=\beta+i\gamma\colon\beta>\sigma,~{}|\gamma|\leq
T,~{}L(\rho,\chi)=0\\},$

then

(4.10) $\displaystyle N_{\chi}(1-\eta,T)$
$\displaystyle\leq\frac{\xi^{3}}{200\sqrt{\xi^{2}-1}}\eta^{2}M_{\xi\eta}e^{2.672\,32M_{\xi\eta}}\int_{-T}^{T}\Big{(}\int_{N_{\xi\eta}}^{N_{\xi\eta}^{*}}\Big{|}\sum_{N_{\xi\eta}<p\leq
u}\frac{\chi(p)\log
p}{p^{1+i\tau}}\Big{|}^{2}\frac{du}{u}\Big{)}\sum_{\rho}\Upphi_{\rho_{0},\xi\eta}(\tau)d\tau$
$\displaystyle\leq
1.613\eta^{2}M_{\xi\eta}^{2}e^{2.672\,32M_{\xi\eta}}\int_{-T}^{T}\Big{(}\int_{N_{\xi\eta}}^{N_{\xi\eta}^{*}}\Big{|}\sum_{N_{\xi\eta}<p\leq
u}\frac{\chi(p)\log p}{p^{1+i\tau}}\Big{|}^{2}\frac{du}{u}\Big{)}d\tau$
$\displaystyle=1.613\eta^{2}M_{\xi\eta}^{2}e^{2.672\,32M_{\xi\eta}}\int_{N_{\xi\eta}}^{N_{\xi\eta}^{*}}\Big{(}\int_{-T}^{T}\Big{|}\sum_{N_{\xi\eta}<p\leq
u}\frac{\chi(p)\log p}{p^{1+i\tau}}\Big{|}^{2}d\tau\Big{)}\frac{du}{u}.$

##  5\. Proof of Theorem 1.2

We begin by summing (4.10) over all primitive Dirichlet characters with
modulus at most $Q$. In keeping with [3, 8], we will choose $T=Q$, so (x)
becomes the condition $Q>10^{4}$. We record

(5.1) $\displaystyle\mathcal{N}_{\xi\eta}$
$\displaystyle=A\eta\Big{(}\frac{4}{3}\log Q+13.04\Big{)},$ $\displaystyle
M_{\xi\eta}$ $\displaystyle=(\alpha-1)\mathcal{N}_{\eta}\geq 63.925,$
$\displaystyle N_{\xi\eta}$
$\displaystyle=\exp\Big{(}\frac{A_{0}M_{\xi\eta}}{\xi\eta}\Big{)}>10^{106}Q^{3.175\,142}.$

###  5.1. Proof when $Q\leq 10^{4}$

Since $\sigma\geq\frac{39}{40}$ in Theorem 1.2, this follows immediately from
Lemma 2.2.

###  5.2. Bound for $N(\sigma,Q)$ when $Q>10^{4}$

We sum (4.10) over all primitive $\chi\,(\mathrm{mod}\,\,q_{\chi})$ with
$q_{\chi}\leq Q$, leading us to now bound

(5.2) $\displaystyle\sum_{q\leq
Q}\sum_{\begin{subarray}{c}\chi\,(\mathrm{mod}\,\,q)\\\ \textup{$\chi$
primitive}\end{subarray}}N_{\chi}(1-\eta,T)$ $\displaystyle\leq
1.613\eta^{2}M_{\xi\eta}^{2}e^{2.672\,32M_{\xi\eta}}\int_{N_{\xi\eta}}^{N_{\xi\eta}^{*}}\Big{(}\sum_{q\leq
Q}\sum_{\begin{subarray}{c}\chi\,(\mathrm{mod}\,\,q)\\\ \textup{$\chi$
primitive}\end{subarray}}\int_{-T}^{T}\Big{|}\sum_{N_{\xi\eta}<p\leq
u}\frac{\chi(p)\log p}{p^{1+i\tau}}\Big{|}^{2}d\tau\Big{)}\frac{du}{u}.$

To proceed, we cite a “pre-sifted” version of the large sieve inequality for
Dirichlet characters.

######  Lemma 5.1.

Let $(a_{n})_{n=1}^{\infty}$ be a sequence of complex numbers. Let
$P^{-}(n)=\min\\{p\colon p|n\\}$ if $n>1$ and $P^{-}(1)=\infty$. If $Q\geq 1$,
$U\geq 0$, and $V\geq 1$, then

$\sum_{q\leq
Q}\Big{(}\log\frac{Q}{q}\Big{)}\sum_{\begin{subarray}{c}\chi\,(\mathrm{mod}\,\,q)\\\
\textup{$\chi$
primitive}\end{subarray}}\Big{|}\sum_{\begin{subarray}{c}U<n\leq U+\lfloor
V\rfloor\\\
P^{-}(n)>Q\end{subarray}}a_{n}\chi(n)\Big{|}^{2}\leq(V+Q^{2}-1)\sum_{\begin{subarray}{c}U<n\leq
U+\lfloor V\rfloor\\\ P^{-}(n)>Q\end{subarray}}|a_{n}|^{2}.$

###### Proof.

This is [7, Theorem 9.11], a numerically optimal version of [8, Lemma 4]. ∎

######  Corollary 5.2.

Let $(a_{n})_{n=1}^{\infty}$ be a sequence of complex numbers such that
$\sum_{n=1}^{\infty}|a_{n}|$ converges. If $Q\geq 1$ and $T\geq 1$, then

$\sum_{q\leq
Q}\Big{(}\log\frac{Q}{q}\Big{)}\sum_{\begin{subarray}{c}\chi\,(\mathrm{mod}\,\,q)\\\
\textup{$\chi$
primitive}\end{subarray}}\int_{-T}^{T}\Big{|}\sum_{\begin{subarray}{c}n\geq
1\\\ P^{-}(n)>Q\end{subarray}}a_{n}\chi(n)n^{-it}\Big{|}^{2}dt\leq
7\sum_{\begin{subarray}{c}n\geq 1\\\
P^{-}(n)>Q\end{subarray}}|a_{n}|^{2}(n+Q^{2}\max\\{T,3\\}).$

###### Proof.

We proceed exactly as in the proof of [28, Corollary 6.3], with the exception
that we use Lemma 5.1 instead of the classical large sieve inequality (as
stated as [28, Lemma 6.1]). ∎

Being only slightly inefficient, we obtain from Corollary 5.2 the bound

$\displaystyle\sum_{q\leq
Q}\sum_{\begin{subarray}{c}\chi\,(\mathrm{mod}\,\,q)\\\ \textup{$\chi$
primitive}\end{subarray}}\int_{-T}^{T}\Big{|}\sum_{\begin{subarray}{c}n\geq
1\\\ P^{-}(n)>Q^{1.087\,571}\end{subarray}}a_{n}\chi(n)n^{-it}\Big{|}^{2}dt$
$\displaystyle\leq\frac{11.42}{\log Q}\sum_{q\leq
Q^{1.087\,571}}\Big{(}\log\frac{Q^{1.087\,571}}{q}\Big{)}\sum_{\begin{subarray}{c}\chi\,(\mathrm{mod}\,\,q)\\\
\textup{$\chi$
primitive}\end{subarray}}\int_{-T}^{T}\Big{|}\sum_{\begin{subarray}{c}n\geq
1\\\ P^{-}(n)>Q^{1.087\,571}\end{subarray}}a_{n}\chi(n)n^{-it}\Big{|}^{2}dt$
$\displaystyle\leq\frac{79.94}{\log Q}\sum_{\begin{subarray}{c}n\geq 1\\\
P^{-}(n)>Q^{1.087\,571}\end{subarray}}|a_{n}|^{2}(n+Q^{2.175\,142}\max\\{T,3\\}).$

Fix $u\in[N_{\xi\eta},N_{\xi\eta}^{*}]$. By (5.1), we have that
$Q^{1.087\,571}<N_{\xi\eta}$. Thus, if we insert $T=Q>10^{4}$ and let

$a_{n}=\begin{cases}(\log n)/n&\mbox{if $n\in[N_{\xi\eta},u]$ is prime,}\\\
0&\mbox{otherwise,}\end{cases}$

then

$\sum_{q\leq Q}\sum_{\begin{subarray}{c}\chi\,(\mathrm{mod}\,\,q)\\\
\textup{$\chi$
primitive}\end{subarray}}\int_{-Q}^{Q}\Big{|}\sum_{N_{\xi\eta}<p\leq
u}\frac{\chi(p)\log p}{p^{1+i\tau}}\Big{|}^{2}d\tau\leq\frac{79.94}{\log
Q}\sum_{N_{\xi\eta}<p\leq u}\frac{(\log
p)^{2}}{p}\Big{(}1+\frac{Q^{3.175\,142}}{p}\Big{)}.$

By (5.1), we find that if $p>N_{\xi\eta}$, then $Q^{3.175\,142}/p\leq
10^{-106}$. It follows that

$\sum_{q\leq Q}\sum_{\begin{subarray}{c}\chi\,(\mathrm{mod}\,\,q)\\\
\textup{$\chi$
primitive}\end{subarray}}\int_{-Q}^{Q}\Big{|}\sum_{N_{\xi\eta}<p\leq
u}\frac{\chi(p)\log p}{p^{1+i\tau}}\Big{|}^{2}d\tau\leq\frac{80}{\log
Q}\sum_{N_{\xi\eta}<p\leq u}\frac{(\log p)^{2}}{p}.$

For $x\geq 2$, it follows from [30, (3.24)] and [23] that $\log
x-2\leq\sum_{p\leq x}\frac{\log p}{p}\leq\log x$. Partial summation yields
$\sum_{x<p\leq y}\frac{(\log p)^{2}}{p}\leq\frac{1}{2}((\log y)^{2}+4\log
y-(\log x)^{2})$, so

(5.3) $\sum_{q\leq Q}\sum_{\begin{subarray}{c}\chi\,(\mathrm{mod}\,\,q)\\\
\textup{$\chi$
primitive}\end{subarray}}\int_{-Q}^{Q}\Big{|}\sum_{N_{\xi\eta}<p\leq
u}\frac{\chi(p)\log p}{p^{1+i\tau}}\Big{|}^{2}d\tau\leq\frac{40}{\log Q}((\log
u)^{2}+4\log u-(\log N_{\xi\eta})^{2}).$

From (5.1), (5.2), and (5.3), we deduce the bound (recall (1.2))

$\displaystyle N(1-\eta,Q)$ $\displaystyle=\sum_{q\leq
Q}\sum_{\begin{subarray}{c}\chi\,(\mathrm{mod}\,\,q)\\\ \textup{$\chi$
primitive}\end{subarray}}N_{\chi}(1-\eta,Q)$ $\displaystyle\leq
1.613\eta^{2}M_{\xi\eta}^{2}e^{2.672\,32M_{\xi\eta}}\int_{N_{\xi\eta}}^{N_{\xi\eta}^{*}}\Big{(}\sum_{q\leq
Q}\sum_{\begin{subarray}{c}\chi\,(\mathrm{mod}\,\,q)\\\ \textup{$\chi$
primitive}\end{subarray}}\int_{-Q}^{Q}\Big{|}\sum_{N_{\xi\eta}<p\leq
u}\frac{\chi(p)\log p}{p^{1+i\tau}}\Big{|}^{2}d\tau\Big{)}\frac{du}{u}$
$\displaystyle\leq\frac{64.52\eta^{2}M_{\xi\eta}^{2}e^{2.672\,32M_{\xi\eta}}}{\log
Q}\int_{N_{\xi\eta}}^{N_{\xi\eta}^{*}}((\log u)^{2}+4\log u-(\log
N_{\xi\eta})^{2})\frac{du}{u}$ $\displaystyle\leq
27\,034\frac{M_{\xi\eta}^{5}}{\eta\log Q}e^{2.672\,32M_{\xi\eta}}.$

Since $T=Q>10^{4}$, it follows from (5.1) and (vii) that $63.925\leq
M_{\xi\eta}\leq 1151\eta\log Q$ and

$N(1-\eta,Q)\leq 31\,116\,134M_{\xi\eta}^{4}e^{2.672\,32M_{\xi\eta}}\leq
6.441\,034\times 10^{12}e^{2.741M_{\xi\eta}}\leq 1.180\,016\times
10^{87}(e^{968.1455}Q^{99})^{\eta}.$

Since $Q=T$, we have that $\mathcal{L}=\log(QT)=2\log Q$. Therefore, using
(vii), when we choose $\sigma=1-\eta$, we arrive at

$N(\sigma,Q)\leq 1.180\,016\times
10^{87}(e^{968.1455}Q^{99})^{1-\sigma},\qquad 1-\frac{1}{10A}\leq\sigma\leq
1-\frac{1}{6A\log Q}.$

If $1-\frac{1}{6A\log Q}<\sigma\leq 1$, then by Theorem 2.6, we have that

$N(\sigma,Q)\leq 1\leq 1.180\,016\times
10^{87}(e^{968.1455}Q^{99})^{1-\sigma}.$

We have therefore proved that

(5.4) $N(\sigma,Q)\leq 1.180\,016\times
10^{87}(e^{968.1455}Q^{99})^{1-\sigma},\qquad\sigma\geq 1-\frac{1}{10A}.$

This is slightly better than the result stated in Theorem 1.2.

###  5.3. Bound for $N^{*}(\sigma,Q)$ when $Q>10^{4}$

It follows from Theorem 2.6 that our claimed bound is trivial when $Q\leq
400\,000$, so we assume otherwise. If

$\sigma>1-\frac{\log(\frac{1}{(1-\beta_{1}(Q))(670\,564.676+454\,935.78\log
Q)})}{104.645+70.996\log Q},$

then by Theorem 2.15 with $T=Q>400\,000$, we must have that
$N^{*}(\sigma,Q)=0$. If

(5.5) $\frac{1}{2}<\sigma\leq
1-\frac{\log(\frac{1}{(1-\beta_{1}(Q))(670\,564.676+454\,935.78\log
Q)})}{104.645+70.996\log Q},$

then upon rearrangement and applying our bound $Q>400\,000$, we must have

(5.6) $1\leq 506\,921(1-\beta_{1}(Q))(\log Q)(e^{104.645}Q^{71})^{1-\sigma}.$

Thus, if $\sigma\geq 1-\frac{1}{10A}$ satisfies (5.5) and $Q>400\,000$, then
by (5.4) and (5.6), $N^{*}(\sigma,Q)$ is at most

$506\,921(1-\beta_{1}(Q))(\log Q)(e^{104.645}Q^{71})^{1-\sigma}N(\sigma,Q)\leq
6\times 10^{92}(1-\beta_{1}(Q))(\log Q)(10^{466}Q^{170})^{1-\sigma}.$

If $(1-\beta_{1}(Q))\log Q$ is at least 1, then it follows from our bound for
$N(\sigma,Q)$ that we can replace $(1-\beta_{1}(Q))\log Q$ with
$\min\\{1,(1-\beta_{1}(Q))\log Q\\}$. Again, this is slightly better than the
result stated in Theorem 1.2.

##  6\. An explicit power savings in Sárközy’s theorem

In this section, we prove Theorem 1.1. For ease of comparison, we will adhere
as closely as possible to the setup and notation of Green [10] and make
explicit his key results on zeros of Dirichlet $L$-functions. To this end, we
state the following corollary of Theorem 1.2.

######  Corollary 6.1.

Let $Q\geq 3$ and $\sigma\geq 0$. With $N(\sigma,Q)$ and $N^{*}(\sigma,Q)$ as
in (1.2), there holds

$N(\sigma,Q)\leq 10^{88}(10^{421}Q^{127})^{1-\sigma},\qquad
N^{*}(\sigma,Q)\leq 10^{93}\min\\{1,(1-\beta_{1}(Q))\log
Q\\}(10^{466}Q^{198})^{1-\sigma}.$

###### Proof.

When $\sigma\geq\frac{39}{40}$, Corollary 6.1 is weaker than Theorem 1.2. When
$0\leq\sigma<\frac{39}{40}$, we consider two cases. If $Q\geq 10^{4}$, then
the inflated exponents make our estimates weaker than what one can deduce from
Lemma 2.8 and the estimate $1-\beta_{1}(Q)\geq 100/(\sqrt{Q}(\log Q)^{2})$
from Theorem 2.6. If $Q<10^{4}$, then the total number of zeros that need to
be counted is bounded above by $N(0,10^{4})$. Our upper bound for
$N(\sigma,Q)$ when $0\leq\sigma<\frac{39}{40}$ and $Q<10^{4}$ greatly exceeds
the bound for $N(0,10^{4})$ that follows from Lemma 2.8 and Theorem 2.6 again.
∎

Define

$\lambda_{1}=\frac{1}{20},\qquad\lambda_{2}=10^{103},\qquad\lambda_{3}=198,\qquad
M=7\times 3^{6}\times 2^{22\,993},$

which correspond to explicit choices of the constants in [10, Section 7]. It
is important to note that [10] adopts the non-standard notation of writing
nontrivial zeros $\rho$ of $L(s,\chi)$ as $\beta+i\gamma=1-\sigma+i\gamma$. In
order to avoid confusion with notation in the rest of this paper while making
the comparison with [10] straightforward, we write most of the results in this
section in terms of $1-\beta$ instead of $\beta$. We record an explicit zero
free region via two propositions.

######  Proposition 6.2.

Let $Q\geq 10$ and $\mathscr{L}(s,Q)$ be as in (1.1). With possibly one
exception, say $\rho_{*}$, every zero $\rho=\beta+i\gamma$ of
$\mathscr{L}(s,Q)$ satisfies $1-\beta\geq\frac{\lambda_{1}}{\log Q}$. If the
exceptional zero $\rho_{*}$ exists, then there exists a unique primitive
quadratic character $\chi_{*}\,(\mathrm{mod}\,\,q_{*})$ with $q_{*}\leq Q$
such that $\rho_{*}$ is a real and simple zero of $L(s,\chi_{*})$. The zero
$\rho_{*}$, if it exists, is called the exceptional zero at scale $Q$.

###### Proof.

This explicit version of [10, Proposition 7.1] follows from Theorem 2.6. ∎

Second, we record two explicit results on log-free zero density estimates.

######  Proposition 6.3.

Let $Q\geq 10$. If $\alpha\geq 0$, then

$\sum_{q\leq Q}~{}\sum_{\begin{subarray}{c}\chi\,(\mathrm{mod}\,\,q)\\\
\textup{$\chi$
primitive}\end{subarray}}~{}\sum_{\begin{subarray}{c}\rho=\beta+i\gamma\\\
\beta\geq\alpha\\\ |\gamma|\leq
Q\end{subarray}}1\leq\lambda_{2}Q^{\lambda_{3}(1-\alpha)}.$

###### Proof.

This explicit version of [10, Proposition 7.4] follows from Corollary 6.1. ∎

######  Proposition 6.4.

Let $Q\geq 10$. If $\alpha\geq 0$ and there is an exceptional zero $\rho_{*}$
at scale $Q$, then

$\sum_{q\leq Q}~{}\sum_{\begin{subarray}{c}\chi\,(\mathrm{mod}\,\,q)\\\
\textup{$\chi$
primitive}\end{subarray}}~{}\sum_{\begin{subarray}{c}\rho=\beta+i\gamma\neq\rho_{*}\\\
\beta\geq\alpha\\\ |\gamma|\leq
Q\end{subarray}}1\leq\lambda_{2}(1-\rho_{*})(\log
Q)Q^{\lambda_{3}(1-\alpha)}.$

###### Proof.

This explicit version of [10, Proposition 7.5] follows from Corollary 6.1. ∎

These preliminary results establish a helpful technical estimate.

######  Proposition 6.5.

Set $B=323\,905$. The following holds for any parameter $Q\geq 10$.

  * •

If there is no exceptional zero at scale $Q$, then

$4M\sum_{q\leq Q}~{}\sum_{\begin{subarray}{c}\chi\,(\mathrm{mod}\,\,q)\\\
\textup{$\chi$
primitive}\end{subarray}}~{}\sum_{\begin{subarray}{c}\rho=\beta+i\gamma\\\
\beta\geq 1/2\\\ |\gamma|\leq Q\end{subarray}}Q^{-B(1-\beta)}\leq 1.$

  * •

If there is an exceptional zero $\rho_{*}$ at scale $Q$, then

$Q^{-B(1-\rho_{*})}+2M\sum_{q\leq
Q}~{}\sum_{\begin{subarray}{c}\chi\,(\mathrm{mod}\,\,q)\\\ \textup{$\chi$
primitive}\end{subarray}}~{}\sum_{\begin{subarray}{c}\rho=\beta+i\gamma\neq\rho_{*}\\\
\beta\geq 1/2\\\ |\gamma|\leq Q\end{subarray}}Q^{-B(1-\beta)}\leq 1.$

###### Proof.

This result parallels [10, Proposition 7.6]. Assume there is no exceptional
zero at scale $Q$. By partial summation and Proposition 6.3, we obtain the
bound

$\displaystyle\sum_{q\leq
Q}~{}\sum_{\begin{subarray}{c}\chi\,(\mathrm{mod}\,\,q)\\\ \textup{$\chi$
primitive}\end{subarray}}~{}\sum_{\begin{subarray}{c}\rho=\beta+i\gamma\\\
\beta\geq 1/2\\\ |\gamma|\leq Q\end{subarray}}Q^{-B(1-\beta)}$
$\displaystyle\leq\lambda_{2}(e^{-\lambda_{1}(B-\lambda_{3})}-Q^{-(B-\lambda_{3})/2})+\lambda_{2}B(\log
Q)\int_{1/2}^{1-\lambda_{1}/\log Q}Q^{-(B-\lambda_{3})(1-\sigma)}d\sigma$
$\displaystyle=\frac{\lambda_{2}(2B-\lambda_{3})}{B-\lambda_{3}}(e^{-\lambda_{1}(B-\lambda_{3})}-Q^{-(B-\lambda_{3})/2})\leq
1.24\times 10^{-6926}<1.28\times 10^{-6926}\leq\frac{1}{4M}.$

If there is an exceptional zero $\rho_{*}$ at scale $Q$, then a similar
calculation with Proposition 6.4 shows

$\displaystyle Q^{-B(1-\rho_{*})}+2M\sum_{q\leq
Q}~{}\sum_{\begin{subarray}{c}\chi\,(\mathrm{mod}\,\,q)\\\ \textup{$\chi$
primitive}\end{subarray}}~{}\sum_{\begin{subarray}{c}\rho=\beta+i\gamma\neq\rho_{*}\\\
\beta\geq 1/2\\\ |\gamma|\leq Q\end{subarray}}Q^{-B(1-\beta)}$
$\displaystyle\leq Q^{-B(1-\rho_{*})}+\frac{1}{2}(1-\rho_{*})\log Q.$

Note that $B$ satisfies $e^{-Bx}+\frac{1}{2}x\leq 1$ for
$0<x\leq\lambda_{1}=\frac{1}{20}$. Thus, when $x=(1-\rho_{*})\log Q$ in this
exceptional case, the desired result follows. ∎

This leads to an explicit version of a key technical result required by Green.

######  Proposition 6.6.

Set $c=9\times 10^{-14}$. Suppose that $\sigma_{0}\leq\frac{1}{8}$. Let $Q\geq
10$, and let $N\geq Q^{1/c}$ be a large integer. There exists
$T\in[N^{c},N^{1/120}]$, such that at least one of the following holds.

  1. (1)

We have that

$\sum_{q\leq T}~{}\sum_{\begin{subarray}{c}\chi\,(\mathrm{mod}\,\,q)\\\
\textup{$\chi$
primitive}\end{subarray}}~{}\sum_{\begin{subarray}{c}\rho=\beta+i\gamma\\\
\frac{1}{2}\leq\beta\leq 1-\sigma_{0}\\\ |\gamma|\leq
T\end{subarray}}N^{-\frac{1}{16}(1-\beta)}\leq\frac{1}{2M}.$

  2. (2)

Let $\rho_{1}=\beta_{1}+i\gamma_{1}$ be a zero belonging to a character with
conductor $q_{1}\leq T$ and $|\gamma_{1}|\leq T$ such that $\beta_{1}$ is
maximal. Then $\rho_{1}$ is a simple, real zero of a real character $\chi_{1}$
of conductor $q_{1}\leq T^{1/10^{6}}$, and

$N^{-\frac{1}{16}(1-\beta_{1})}+2M\sum_{q\leq
T}~{}\sum_{\begin{subarray}{c}\chi\,(\mathrm{mod}\,\,q)\\\ \textup{$\chi$
primitive}\end{subarray}}~{}\sum_{\begin{subarray}{c}\rho=\beta+i\gamma\neq\rho_{*}\\\
1/2\leq\beta\leq 1-\sigma_{0}\\\ |\gamma|\leq
T\end{subarray}}N^{-\frac{1}{16}(1-\beta)}\leq 1.$

Also, regardless of whether an exceptional zero exists, we have the bound

$\sum_{q\leq Q}~{}\sum_{\begin{subarray}{c}\chi\,(\mathrm{mod}\,\,q)\\\
\textup{$\chi$
primitive}\end{subarray}}~{}\sum_{\begin{subarray}{c}\rho=\beta+i\gamma\\\
\beta\geq 1/2\\\ |\gamma|\leq T\end{subarray}}1\ll T.$

###### Proof.

For (1) and (2), we have that
$c<\min(\frac{1}{16B},\frac{10^{-6}\lambda_{1}}{32\log(4M)})=9.79\ldots\times
10^{-14}$, so the proof is identical to [10, Proposition 7.7] with our
Propositions 6.2-6.5 replacing [10, Propositions 7.1, 7.4, 7.5, and 7.6]. The
last bound follows from Jutila’s estimate (1.4) with $\sigma\geq
1-\frac{1}{8}$ and $\varepsilon=\frac{1}{100}$. ∎

Finally, with all the necessary explicit results in hand, we may prove Theorem
1.1.

###### Proof of Theorem 1.1.

In [10, Theorem 2.2], Green proves the existence of constants
$0<\kappa_{2}<\kappa_{1}$ such that if $A\subseteq\\{1,\ldots,N\\}$ satisfies
the hypotheses of Theorem 1.1, then $|A|\ll N^{1-(\kappa_{1}-\kappa_{2})}$. By
the discussion in [10, Section 15.1], there exists a constant $c>0$ such that
if $\tau_{0}\in[c,\frac{1}{120}]$ and $T=N^{\tau_{0}}$ in [10, Proposition
7.7], then we may take $\kappa_{1}=\frac{\tau_{0}}{6\times 10^{4}}$ and
$\kappa_{2}=\frac{2\tau_{0}}{10^{6}}$. (In [10, Section 15.1], these are
written as $\kappa_{1}=\frac{1}{6}c_{2}\tau_{0}$ and
$\kappa_{2}=2c_{3}\tau_{0}$, where $c_{i}=(\frac{1}{100})^{i}$.) Since our
Proposition 6.6 makes [10, Proposition 7.7] explicit, we may take $c=9\times
10^{-14}$, in which case

$\kappa_{1}-\kappa_{2}=\tau_{0}\Big{(}\frac{1}{6\times
10^{4}}-\frac{2}{10^{6}}\Big{)}\geq\frac{9}{10^{14}}\Big{(}\frac{1}{6\times
10^{4}}-\frac{2}{10^{6}}\Big{)}=\frac{1.32}{10^{18}},$

as desired. ∎

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