abconsymbol=c abcon2symbol=C

# Refinements to the prime number theorem for arithmetic progressions

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

We prove a version of the prime number theorem for arithmetic progressions
that is uniform enough to deduce the Siegel–Walfisz theorem, Hoheisel’s
asymptotic for intervals of length $x^{1-\delta}$, a Brun–Titchmarsh bound,
and Linnik’s bound on the least prime in an arithmetic progression as
corollaries. Our proof uses the Vinogradov–Korobov zero-free region and a
refinement of Bombieri’s “repulsive” log-free zero density estimate.
Improvements exist when the modulus is sufficiently powerful.

##  1\. Introduction and statement of results

###  1.1. Main result

Let $q\geq 2$. The prime number theorem for arithmetic progressions states
that as $x\to\infty$, the primes $p\leq x$ equidistribute in the $\varphi(q)$
residue classes $a\,(\mathrm{mod}{\,q})$ with $\gcd(a,q)=1$. Under the
generalized Riemann hypothesis (GRH), primes in intervals of length $h$
equidistribute in residue classes modulo $q$ once
$h/\varphi(q)\geq\sqrt{x}(\log x)^{2}$; that is,

(1.1) $\sum_{\begin{subarray}{c}x-h<p\leq x\\\ p\equiv
a\,(\mathrm{mod}{\,q})\end{subarray}}\log
p=\frac{h}{\varphi(q)}+O(\sqrt{x}(\log x)^{2}),\qquad x\geq q.$

Unconditional progress towards (1.1) has classical origins; the latest
benchmarks are described in Section 1.2. Each such result usually addresses a
specific feature: long intervals (e.g., $h=x$), short intervals (e.g.,
$h=x^{1-\delta}$), small moduli (e.g., $q\leq(\log x)^{100}$), or large moduli
(e.g., $q\leq x^{\delta}$). We prove a version of the prime number theorem for
arithmetic progressions that interpolates these benchmarks, giving the most
uniform progress towards (1.1) that current results on Dirichlet $L$-functions
permit. A major obstacle to uniformity is the possible existence of real zeros
of Dirichlet $L$-functions near $s=1$. McCurley [14] proved that
$\prod_{\chi\,(\mathrm{mod}{\,q})}L(s,\chi)$ has at most one real zero
$\beta_{1}\geq 1-\frac{1}{50\log q}$. If this exceptional zero $\beta_{1}$
exists, then it is simple, and there exists a unique nontrivial real Dirichlet
character $\chi_{1}\,(\mathrm{mod}{\,q})$ such that $L(\beta_{1},\chi_{1})=0$.
We paraphrase this with “$(\beta_{1},\chi_{1})$ exists”.

######  Theorem 1.1.

Let $q\geq 2$ and $a$ be coprime integers, and let $4\leq h\leq x$. Define
$\lambda=\lambda(x,q,a,h)$ and $\theta$ by

$\lambda:=\begin{cases}\displaystyle\frac{1}{h}\int_{x-h}^{x}(1-\chi_{1}(a)t^{\beta_{1}-1})dt&\mbox{if
$(\beta_{1},\chi_{1})$ exists,}\\\
1&\mbox{otherwise}\end{cases}\quad\text{and}\quad\theta:=\begin{cases}71/75&\mbox{if
$(\beta_{1},\chi_{1})$ exists,}\\\ 7/12&\mbox{otherwise.}\end{cases}$

For all $0<\varepsilon<1-\theta$, there is a constant
$\Cl[abcon]{main}=\Cr{main}(\varepsilon)>0$ such that if $\lambda
h/\varphi(q)\geq x^{\theta+\varepsilon}$, then

$\sum_{\begin{subarray}{c}x-h<p\leq x\\\ p\equiv
a\,(\mathrm{mod}{\,q})\end{subarray}}\log p=\frac{\lambda
h}{\varphi(q)}\Big{[}1+O_{\varepsilon}\Big{(}\exp\Big{(}\frac{-\Cr{main}\log
x}{\log
q+(\log\frac{x}{h})^{\frac{2}{3}}(\log^{+}\log\frac{x}{h})^{\frac{1}{3}}+(\log
x)^{\frac{2}{5}}(\log\log x)^{\frac{1}{5}}}\Big{)}\Big{)}\Big{]}.$

The implied constant and $\Cr{main}$ are effectively computable, and
$\log^{+}(u)=\max\\{0,\log u\\}$.

######  Remark 1.2.

Let $A\geq 1$. If $(\beta_{1},\chi_{1})$ exists and $q\leq(\log x)^{A}$, then
Theorem 1.1 holds with $\theta=\frac{7}{12}$, but $\Cr{main}$ will depend
ineffectively on $A$. We justify this in Section 4 using Siegel’s theorem [2,
Ch. 21]: For all $\varepsilon>0$, there exists a constant $b_{\varepsilon}>0$
such that

(1.2) $1-\beta_{1}\geq b_{\varepsilon}q^{-\varepsilon}.$

The constant $b_{\varepsilon}$ is effectively computable when
$\varepsilon\geq\frac{1}{2}$ [18, Thm 3].

######  Remark 1.3.

If there is a constant $0<\Cl[abcon]{relative}\leq\frac{1}{50}$, independent
of $q$, such that $\beta_{1}=1-\frac{\Cr{relative}}{\log q}$, then one may
apply Theorem 1.1 with $\lambda=1$ and $\theta=\frac{7}{12}$. The implied
constant and $\Cr{main}$ will depend effectively on $\Cr{relative}$. We
justify these assertions in Section 4.

######  Remark 1.4.

The fraction $\frac{71}{75}=0.94\bar{6}$ conveniently approximates
$0.9465267\ldots$ Without assuming bounds for Dirichlet $L$-functions beyond
the recently proven hybrid Weyl bound [17], this appears to be the limit of
our method when $(\beta_{1},\chi_{1})$ exists. See Remark 5.8.

If $(\beta_{1},\chi_{1})$ does not exist, then $\lambda=1$ and $h/\varphi(q)$
is the main term in Theorem 1.1. If $(\beta_{1},\chi_{1})$ exists, then the
notion of a “main term” and an “error term” in Theorem 1.1 depends on how one
views the role of $\beta_{1}$. We suggest the perspective of viewing

(1.3) $\frac{\lambda
h}{\varphi(q)}=\frac{h}{\varphi(q)}-\frac{\chi_{1}(a)}{\varphi(q)}\frac{x^{\beta_{1}}-(x-h)^{\beta_{1}}}{\beta_{1}}$

as the sum of a “primary main term” and a “secondary main term”, leaving

(1.4) $\frac{\lambda h}{\varphi(q)}\exp\Big{(}\frac{-\Cr{main}\log x}{\log
q+(\log\frac{x}{h})^{\frac{2}{3}}(\log^{+}\log\frac{x}{h})^{\frac{1}{3}}+(\log
x)^{\frac{2}{5}}(\log\log x)^{\frac{1}{5}}}\Big{)}$

as an “error term”. This perspective is implicit in Linnik’s work on the least
prime $p\equiv a\,(\mathrm{mod}{\,q})$ [12]. In light of our piecewise
definition of $\lambda$, this perspective accommodates all ranges of $q$ and
$x$ under our consideration, regardless of whether $(\beta_{1},\chi_{1})$
exists.

If $(\beta_{1},\chi_{1})$ exists, $x^{1/2}\leq h\leq x$, and $x\geq 16$, then
there exists $\xi\in(x,x+h)$ such that

(1.5) $\lambda=1-\chi_{1}(a)\xi^{\beta_{1}-1}.$

Furthermore, if there exists $A\geq 1$ such that $q\leq(\log x)^{A}$, then
(1.2) with $\varepsilon=\frac{1}{3A}$ shows that

(1.6) $\lambda=1+O(\exp(-b_{\frac{1}{3A}}(\log x)^{2/3})).$

The $O$-term in (1.6) is readily absorbed by (1.4). When $q$ is larger, we
allow the contribution from $\beta_{1}$ in (1.3) to function as a “secondary
main term” rather than attempt to push the contribution into any sort of
“error term”. This permits Theorem 1.1 to effectively count primes in
arithmetic progressions modulo $q$ as long as the expected number of primes
congruent to $a\,(\mathrm{mod}{\,q})$ in the interval $(x-h,x]$, namely
$\lambda h/\varphi(q)$, exceeds $x^{\theta}$. If, in addition, one has $(\log
x)/(\log q)\to\infty$, then the expected number of primes becomes an effective
asymptotic count, even if $\beta_{1}$ exists. The key point is that (1.3) and
(1.4) are each scaled by $\lambda$, so the existence of $(\beta_{1},\chi_{1})$
proportionally affects both quantities. This differs from many traditional
presentations of the prime number theorem for arithmetic progressions (e.g.,
[2, 11]) where the contribution from $\beta_{1}$ is viewed as belonging in an
“error term” instead of being treated as a “secondary main term.”

Even if $(\beta_{1},\chi_{1})$ exists, we can always effectively estimate
$\lambda$ using the effective upper bound $\beta_{1}\leq 1-b_{1/2}q^{-1/2}$
from (1.2) as well as the bounds

(1.7) $\frac{1}{8}\min\\{1,(1-\beta_{1})\log x\\}<\lambda<2,$

which we prove in Section 4. Therefore, even in the most restrictive case, for
all $0<\varepsilon<1-\theta$, there exists an effectively computable constant
$\Cl[abcon]{range_{c}onst}=\Cr{range_{c}onst}(\varepsilon)>0$ such that the
condition $\lambda h/\varphi(q)\geq x^{\theta+\varepsilon}$ in Theorem 1.1 is
satisfied when $x\geq\Cr{range_{c}onst}$ and

$h\geq x^{1-\delta_{1}},\qquad q\leq
x^{\delta_{2}},\qquad\delta_{1}+\frac{3}{2}\delta_{2}\leq
1-\theta-\varepsilon.$

If $(\beta_{1},\chi_{1})$ does not exist, then the factor of $\frac{3}{2}$ can
be replaced by $1$.

Taking $h=x$ and $\varepsilon=\frac{1}{1425}$ in Theorem 1.1, we arrive at the
following result.

######  Corollary 1.5.

Let $q\geq 2$ and $a$ be coprime integers and $x\geq 3$. Let
$\lambda=\lambda(x,q,a,x)$ be as in Theorem 1.1. There exists a constant
$\Cl[abcon]{cor_{l}ong}>0$ such that if $\lambda x/\varphi(q)\geq x^{18/19}$,
then

$\sum_{\begin{subarray}{c}p\leq x\\\ p\equiv
a\,(\mathrm{mod}{\,q})\end{subarray}}\log p=\dfrac{\lambda
x}{\varphi(q)}\Big{[}1+O\Big{(}\exp\Big{(}-\Cr{cor_{l}ong}\frac{\log x}{\log
q}\Big{)}+\exp\Big{(}-\Cr{cor_{l}ong}\frac{(\log x)^{3/5}}{(\log\log
x)^{1/5}}\Big{)}\Big{)}\Big{]}.$

The implied constant and $\Cr{cor_{l}ong}$ are absolute and effectively
computable.

######  Remark 1.6.

If $(\beta_{1},\chi_{1})$ exists, then
$\lambda(x,q,a,x)=1-\chi_{1}(a)x^{\beta_{1}-1}/\beta_{1}$.

###  1.2. Connections with classical results

To display the uniformity of Theorem 1.1, we deduce the Siegel–Walfisz theorem
with the best known error term, Hoheisel’s asymptotic prime number theorem for
short intervals with the best known error term, a Brun–Titchmarsh type bound
for short intervals, and Linnik’s bound on the least prime in an arithmetic
progression as corollaries of Theorem 1.1, the ineffective bound (1.6), and
the effective bound (1.7). While all of these results can be proved
individually, the point here is that Theorem 1.1 provides a single asymptotic
result that is uniform enough to deduce all of them. These deductions, which
require no information on $\beta_{1}$ except for (1.2) and the effective
computability of $b_{1/2}$, showcase the flexibility of Theorem 1.1 in
handling any combination of large, medium, or small moduli with short, medium,
or long intervals.

If $A\geq 1$, and $q\leq(\log x)^{A}$, then (1.6) and Theorem 1.1 with $h=x$
imply that there exists an ineffective constant $c_{A}>0$ such that

$\sum_{\begin{subarray}{c}p\leq x\\\ p\equiv
a\,(\mathrm{mod}{\,q})\end{subarray}}\log
p=\frac{x}{\varphi(q)}\Big{(}1+O\Big{(}\exp\Big{(}-c_{A}\frac{(\log
x)^{3/5}}{(\log\log x)^{1/5}}\Big{)}\Big{)}\Big{)}.$

This version of the Siegel–Walfisz theorem was recently proved by
Koukoulopoulos [10]. If $\varepsilon>0$ and one adds the condition that $h\geq
x^{7/12+\varepsilon}(\log x)^{A}$, then (1.6), Theorem 1.1, and Remark 1.2
recover the strongest known version of Hoheisel’s short interval prime number
theorem [5] for arithmetic progressions to moduli $q\leq(\log x)^{A}$: There
exists an ineffective constant $c_{A,\varepsilon}>0$ such that

$\sum_{\begin{subarray}{c}x-h<p\leq x\\\ p\equiv
a\,(\mathrm{mod}{\,q})\end{subarray}}\log
p=\frac{h}{\varphi(q)}\Big{(}1+O_{\varepsilon}\Big{(}\exp\Big{(}-c_{A,\varepsilon}\frac{(\log
x)^{1/3}}{(\log\log x)^{1/3}}\Big{)}\Big{)}\Big{)}.$

Consider now the situation where $q$ is large. Applying Theorem 1.1 with
$\varepsilon=\frac{1}{1425}$, we find that if $\lambda h/\varphi(q)\geq
x^{18/19}$ and $(\log x)/(\log q)\to\infty$, then

$\sum_{\begin{subarray}{c}x-h<p\leq x\\\ p\equiv
a\,(\mathrm{mod}{\,q})\end{subarray}}\log p\sim\frac{\lambda h}{\varphi(q)}.$

If $(\beta_{1},\chi_{1})$ exists, then $\lambda$ satisfies (1.7).
Consequently, for all fixed $\delta>0$, there exists an effectively computable
constant $\Cl[abcon]{BT1}=\Cr{BT1}(\delta)>0$ such that if $x\geq
q^{\Cr{BT1}}$, then

$\frac{h}{\varphi(q)}(1-\beta_{1})\log q\ll\sum_{\begin{subarray}{c}x-h<p\leq
x\\\ p\equiv a\,(\mathrm{mod}{\,q})\end{subarray}}\log
p\leq(2+\delta)\frac{h}{\varphi(q)}.$

When $(\beta_{1},\chi_{1})$ does not exist, the $(1-\beta_{1})\log q$ factor
is deleted, and $2+\delta$ becomes $1+\delta$. The upper bound is a weak form
of the Brun–Titchmarsh theorem for short intervals (cf. [16, Theorem 1.2]).
Since $\beta_{1}<1$, the lower bound implies that there exists a prime $p\leq
q^{\Cr{BT1}}$ such that $p\equiv a\,(\mathrm{mod}{\,q})$, as Linnik proved
[12].

### Organization

In Section 2, we state a new version of the prime number theorem for
arithmetic progressions (Theorem 2.2) that depends only on a zero-free region
for $\prod_{\chi\,(\mathrm{mod}{\,q})}L(s,\chi)$. The proof of Theorem 2.2
requires a certain log-free zero density estimate (Theorem 2.1). We prove
Theorem 1.1 assuming these two results and the Vinogradov–Korobov zero-free
regions for Dirichlet $L$-functions. In Section 3, we use Theorem 2.2 and a
zero-free region for $\prod_{\chi\,(\mathrm{mod}{\,q})}L(s,\chi)$ due to
Iwaniec to refine Theorem 1.1 when $q$ is sufficiently “powerful” and provide
an application to primes with prescribed digits. We also compare and contrast
our work with that of Gallagher [3]. We prove Theorem 2.2 in Section 4, and we
prove Theorem 2.1 in Section 5.

### Acknowledgements

We thank Kevin Ford, Andrew Granville, Kannan Soundararajan, and Joni
Teräväinen for their encouragement and helpful conversations.

##  2\. Proof of Theorem 1.1

Let $q\geq 2$, $T\geq 1$ and $0\leq\sigma\leq 1$. For all Dirichlet characters
$\chi\,(\mathrm{mod}{\,q})$, we define

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

If $(\beta_{1},\chi_{1})$ exists, then we define

$N_{\chi_{1}}^{*}(\sigma,T):=|\\{\rho=\beta+i\gamma\colon\beta,\gamma\in\mathbb{R},~{}\beta\geq\sigma,~{}|\gamma|\leq
T,~{}\rho\neq\beta_{1}~{}L(\rho,\chi)=0\\}|.$

Let

$N_{q}(\sigma,T):=\sum_{\chi\,(\mathrm{mod}{\,q})}N_{\chi}(\sigma,T).$

If $(\beta_{1},\chi_{1})$ exists, then let

$N_{q}^{*}(\sigma,T):=\sum_{\begin{subarray}{c}\chi\,(\mathrm{mod}{\,q})\\\
\chi\neq\chi_{1}\end{subarray}}N_{\chi}(\sigma,T)+N_{\chi_{1}}^{*}(\sigma,T).$

Along with the standard theory of Dirichlet $L$-functions [2], our key theorem
is a log-free zero density estimate that encodes the phenomenon that if
$(\beta_{1},\chi_{1})$ exists, then $\beta_{1}$ repels all other zeros of
$\prod_{\chi\,(\mathrm{mod}{\,q})}L(s,\chi)$. We prove the following result in
Section 5.

######  Theorem 2.1.

Let $\varepsilon>0$. Let $q,T\geq 1$ and $0\leq\sigma\leq 1$. We have the
unconditional bound

$N_{q}(\sigma,T)\ll_{\varepsilon}(qT)^{(\frac{12}{5}+\varepsilon)(1-\sigma)}.$

If $(\beta_{1},\chi_{1})$ exists and $\nu(u):=\min\\{1,(1-\beta_{1})\log
u\\}$, then

$N_{q}^{*}(\sigma,T)\ll\nu(qT)(qT)^{\frac{75}{4}(1-\sigma)}.$

######  Remark.

Bombieri [1, Ch. 6] proved that $N_{q}^{*}(\sigma,T)$ is sufficient to deduce
a quantitative form of Deuring–Heilbronn phenomenon wherein $\beta_{1}$ repels
all other zeros of $\prod_{\chi\,(\mathrm{mod}{\,q})}L(s,\chi)$.

Using Theorem 2.1, we prove a highly uniform version of the prime number
theorem for arithmetic progressions that depends only on a zero-free region
for Dirichlet $L$-functions.

######  Theorem 2.2.

Let $q\geq 2$ and $a$ be coprime integers, and let $4\leq h\leq x$. Let
$\lambda$ be as in Theorem 1.1, and define the constant

$\theta:=\begin{cases}71/75&\mbox{if $(\beta_{1},\chi_{1})$ exists,}\\\
7/12&\mbox{otherwise.}\end{cases}$

Let $0<\varepsilon<1-\theta$, and let $\delta:[1,\infty)\to[0,1]$ be a
function such that for all $T\geq e$, the product
$\prod_{\chi\,(\mathrm{mod}{\,q})}L(s,\chi)$ does not vanish in the region

$\\{s\in\mathbb{C}\colon\mathrm{Re}(s)\geq 1-\delta(T),~{}|\mathrm{Im}(s)|\leq
T\\}$

except possibly for $\beta_{1}$, if it exists. If $\lambda h/\varphi(q)\geq
x^{\theta+\varepsilon}$, then

$\sum_{\begin{subarray}{c}x-h<p\leq x\\\ p\equiv
a\,(\mathrm{mod}{\,q})\end{subarray}}\log p=\frac{\lambda
h}{\varphi(q)}\Big{\\{}1+O_{\varepsilon}\Big{(}\sqrt{\frac{x}{h}}\frac{\log
x}{\log
q}\sup\Big{\\{}\frac{x^{-\varepsilon^{2}\delta(t)}}{\sqrt{t}}\colon\frac{ex}{h}\leq
t\leq\frac{x^{(1-\theta)(1-\varepsilon)}}{q}\Big{\\}}+x^{-\varepsilon\theta/2}\Big{)}\Big{\\}}.$

The implied constant is effectively computable and depends at most on
$\varepsilon$.

######  Remark 2.3.

If $(\beta_{1},\chi_{1})$ exists (resp. does not exist) and the exponent
$\frac{75}{4}$ (resp. $\frac{12}{5}$) in Theorem 2.1 is improved to some
constant $\Cl[abcon]{improved_{Z}DE}>0$, then Theorem 2.2 holds with
$\theta=1-\frac{1}{\Cr{improved_{Z}DE}}$.

###### Proof of Theorem 1.1.

We use the Vinogradov–Korobov zero-free region for Dirichlet $L$-functions
[15, Ch. 9] in Theorem 2.2. In particular, there exists an absolute and
effectively computable constant $\Cl[abcon]{ZFR-VK}>0$ such that for $t\geq
e$,

(2.1) $\delta(t)\geq\frac{\Cr{ZFR-VK}}{\log q+(\log t)^{2/3}(\log^{+}\log
t)^{1/3}}.$

Since $e^{-\frac{1}{u+v}}\leq e^{-\frac{1}{2\max\\{u,v\\}}}\leq
e^{-\frac{1}{2u}}+e^{-\frac{1}{2v}}$ for $u,v>0$, it follows that

$\dfrac{x^{-\varepsilon^{2}\delta(t)}}{\sqrt{t}}\leq\exp\Big{(}-\frac{\varepsilon^{2}\Cr{ZFR-
VK}\log x}{2\log q}-\frac{1}{2}\log
t\Big{)}+\exp\Big{(}-\frac{\varepsilon^{2}\Cr{ZFR-VK}\log x}{2(\log
t)^{2/3}(\log^{+}\log t)^{1/3}}-\frac{1}{2}\log t\Big{)}$

for $t\geq e$. We maximize these two exponentials over
$t\in[\frac{ex}{h},\infty)$. The first exponential is maximized at the
leftmost endpoint $t=ex/h$. For the second exponential, notice
$\frac{1}{2}\log t$ and $\frac{\varepsilon^{2}\Cr{ZFR-VK}\log x}{2(\log
t)^{2/3}(\log^{+}\log t)^{1/3}}$ are the same order of magnitude when $\log
t\asymp(\log x)^{3/5}(\log\log x)^{-1/5}$. Thus, the second exponential is
maximized on the interval $[\frac{ex}{h},\infty)$ for a value of $t=t_{0}$
such that either $t_{0}=ex/h$, or $\log t_{0}\asymp_{\varepsilon}(\log
x)^{3/5}(\log\log x)^{-1/5}$ and $t_{0}\geq ex/h$. This implies that

$\displaystyle\sup_{t\geq
ex/h}\Big{\\{}\dfrac{x^{-\varepsilon^{2}\delta(t)}}{\sqrt{t}}\Big{\\}}$
$\displaystyle\leq\sqrt{\frac{h}{x}}\exp\Big{(}-\frac{\varepsilon^{2}\Cr{ZFR-
VK}\log x}{2\log
q}\Big{)}+\sqrt{\frac{h}{x}}\exp\Big{(}-\frac{\varepsilon^{2}\Cr{ZFR-VK}\log
x}{2(\log t_{0})^{2/3}(\log^{+}\log t_{0})^{1/3}}\Big{)}$ $\displaystyle\leq
2\sqrt{\frac{h}{x}}\exp\Big{(}-\frac{\varepsilon^{2}\Cr{ZFR-VK}\log x}{2\log
q+2(\log t_{0})^{2/3}(\log^{+}\log t_{0})^{1/3}}\Big{)},$

since $e^{-\frac{1}{u}}+e^{-\frac{1}{v}}\leq 2e^{-\frac{1}{u+v}}$ for $u,v>0$.
Therefore, by the conditions on $t_{0}$, there exists an effectively
computable constant $\Cr{main}=\Cr{main}({\varepsilon})>0$, depending at most
on $\varepsilon$, such that

(2.2) $\displaystyle\sqrt{\frac{x}{h}}\frac{\log x}{\log
q}\sup\Big{\\{}\frac{x^{-\varepsilon^{2}\delta(t)}}{\sqrt{t}}\colon\frac{ex}{h}\leq
t\leq\frac{x^{(1-\theta)(1-\varepsilon)}}{q}\Big{\\}}+x^{-\varepsilon\theta/2}$
$\displaystyle\leq\frac{2\log x}{\log
q}\exp\Big{(}-\frac{\varepsilon^{2}\Cr{ZFR-VK}\log x}{2\log q+2(\log
t_{0})^{2/3}(\log^{+}\log t_{0})^{1/3}}\Big{)}+x^{-\varepsilon\theta/2}$
$\displaystyle\ll_{\varepsilon}\exp\Big{(}-\frac{\Cr{main}\log x}{\log
q+(\log\frac{x}{h})^{2/3}(\log^{+}\log\frac{x}{h})^{1/3}+(\log
x)^{2/5}(\log\log x)^{1/5}}\Big{)}.$

Inserting this estimate in the conclusion of Theorem 2.2 proves Theorem 1.1. ∎

Notice that the quality of the zero-free region does not impose constraints on
the range of $q$ or the length of the interval $(x-h,x]$. These ranges are
determined only by the log-free zero density estimate in Theorem 2.1. The
zero-free region only influences quality of the $O$-term.

##  3\. Further refinements

Zero-free regions other than (2.1) can be used with Theorem 2.2. In some
situations, this substitution can lead to improvements over Theorem 1.1. We
consider two such examples.

###  3.1. Powerful moduli

Let $q\geq 2$, and let $d=d(q)=\prod_{p|q}p$ be the squarefree part of $q$. It
follows from work of Iwaniec [7] that in Theorem 2.2, we may take

(3.1) $\delta(t)\geq\frac{1/(4\cdot 10^{4})}{\log d+(\log
qt)^{3/4}(\log^{+}\log(qt))^{3/4}}.$

Inserting (3.1) into Theorem 2.2, we immediately obtain another highly uniform
version of the prime number theorem for arithmetic progressions.

######  Corollary 3.1.

Let $q\geq 2$ and $a$ be coprime integers and let $3\leq h<x$. Let
$d=\prod_{p\mid q}p$ be the squarefree part of $q$, let $\lambda$ and $\theta$
be as in Theorem 1.1. For all $\varepsilon>0$, there exists a constant
$\Cl[abcon]{powerful}=\Cr{powerful}(\varepsilon)>0$ such that if $\lambda
h/\varphi(q)\geq x^{\theta+\varepsilon}$ then

$\sum_{\begin{subarray}{c}x-h<p\leq x\\\ p\equiv
a\,(\mathrm{mod}{\,q})\end{subarray}}\log p=\frac{\lambda
h}{\varphi(q)}\Big{[}1+O_{\varepsilon}\Big{(}\exp\Big{(}\frac{-\Cr{powerful}\log
x}{\log
d+(\log\frac{qx}{h})^{\frac{3}{4}}(\log^{+}\log\frac{qx}{h})^{\frac{3}{4}}+(\log
x)^{\frac{3}{7}}(\log\log x)^{\frac{3}{7}}}\Big{)}\Big{)}\Big{]}.$

The implied constant and $\Cr{powerful}$ are effectively computable.

Corollary 3.1 is a substantial improvement over Theorem 1.1 when $q$ is
sufficiently large relative to $d$, in which case $q$ is sufficiently
“powerful.” If $(\beta_{1},\chi_{1})$ exists, then by [21, Lem 6.2], there
exists an absolute and effectively computable constant
$\Cl[abcon]{Swaenepoel}>0$ such that
$\beta_{1}<1-\Cr{Swaenepoel}/(\sqrt{d}(\log d)^{2})$. Consequently, if
$q>\exp(\sqrt{d}(\log d)^{2}/(50\Cr{Swaenepoel}))$, then
$(\beta_{1},\chi_{1})$ does not exist, $\lambda=1$, and if $h/\varphi(q)\geq
x^{7/12+\varepsilon}$, then there exists an effectively computable congmstant
$\Cl[abcon]{powerful2}=\Cr{powerful2}(d)>0$ such that

(3.2) $\sum_{\begin{subarray}{c}x-h<p\leq x\\\ p\equiv
a\,(\mathrm{mod}{\,q})\end{subarray}}\log
p=\frac{h}{\varphi(q)}\Big{[}1+O_{d,\varepsilon}\Big{(}\exp\Big{(}\frac{-\Cr{powerful2}\log
x}{(\log\frac{qx}{h})^{\frac{3}{4}}(\log^{+}\log\frac{qx}{h})^{\frac{3}{4}}+(\log
x)^{\frac{3}{7}}(\log\log x)^{\frac{3}{7}}}\Big{)}\Big{)}\Big{]}.$

This can be used to study primes with prescribed digits. Namely, one can
asymptotically count primes while simultaneously specifying 41.6% of the first
and last digits.

######  Corollary 3.2.

Let $A,B,N\geq 1$ and $\ell\geq 2$ be integers with $A+B<N$. Let
$\mathcal{P}_{\ell}(N)$ be the set of primes with $N$ digits in their base
$\ell$ expansions. For a prime $p\in\mathcal{P}_{\ell}(N)$, write its base
$\ell$ expansion as $\sum_{j=0}^{N-1}d_{j}(p)\ell^{j}$. Fix
$d_{0},d_{1},\ldots,d_{A-1},d_{N-B},d_{N-B+1},\ldots,d_{N-1}\in\\{0,1,\ldots,\ell-1\\}$,
with $d_{N-1}\neq 0$ and $\gcd(d_{1},\ell)=1$. For all
$0<\varepsilon<\frac{5}{12}$, there exists a constant
$\Cl[abcon]{digits}=\Cr{digits}(\varepsilon,\ell)>0$ such that if
$A+B\leq(\frac{5}{12}-\varepsilon)N$, then

$\\#\\{p\in\mathcal{P}_{\ell}(N)\colon\textup{$d_{j}(p)=d_{j}$ for $0\leq
j\leq A-1$ and $N-B+1\leq j\leq N-1$}\\}\\\
=\frac{\ell^{N-A-B}}{\varphi(\ell)}\Big{(}1+O_{\varepsilon,\ell}\Big{(}\exp\Big{(}-\frac{\Cr{digits}N^{1/4}}{(\log
N)^{3/4}}\Big{)}\Big{)}\Big{)}\Big{)}.$

The implied constant and $\Cr{digits}$ are effectively computable.

The proof is nearly immediate from (3.2). The arithmetic progression condition
enables us to prescribe the $A$ least significant digits, while the short
interval condition enables us to prescribe the $B$ most significant digits.
Corollary 3.2 complements recent work of Swaenepoel [21] wherein a small
proportion of the digits can be prescribed without any restriction on how
significant the digits are.

###  3.2. Comparison with work of Gallagher

As far as the authors know, the result closest to Theorem 1.1 in the existing
literature follows from the seminal work of Gallagher [3, Thm 7], which we now
describe using our notation. Recall the definition of $\lambda$ in Theorem
1.1. With the effective bounds in (1.7), Gallagher’s work implies that there
exist absolute and effectively computable constants
$\Cl[abcon]{Gal1},\Cl[abcon]{Gal2}>0$ such that if

(3.3) $x/q\leq h\leq x,\qquad\exp(\sqrt{\log x})\leq q\leq x^{\Cr{Gal1}},$

then

(3.4) $\Big{|}1-\frac{\varphi(q)}{\lambda h}\sum_{\begin{subarray}{c}x-h<p\leq
x\\\ p\equiv a\,(\mathrm{mod}{\,q})\end{subarray}}\log
p\Big{|}\ll\begin{cases}\displaystyle(1-\beta_{1})(\log
q)\exp\Big{(}-\Cr{Gal2}\frac{\log x}{\log q}\Big{)}&\mbox{if
$(\beta_{1},\chi_{1})$ exists,}\vspace{0.1in}\\\
\displaystyle\exp\Big{(}-\Cr{Gal2}\frac{\log x}{\log
q}\Big{)}&\mbox{otherwise.}\end{cases}$

The proof of (3.4) uses a log-free bound on $N_{q}(\sigma,T)$ as in Theorem
2.1, the “standard” zero-free region

$L(\beta+i\gamma,\chi)\neq 0,\qquad\beta+i\gamma\neq\beta_{1},\qquad\beta\leq
1-\frac{\Cl[abcon]{standard_{Z}FR}}{\log(q(|\gamma|+3))},$

and, when $(\beta_{1},\chi_{1})$ exists, the zero-free region

(3.5) $L(\beta+i\gamma,\chi)\neq
0,\qquad\beta+i\gamma\neq\beta_{1},\qquad\beta\leq
1-\Cl[abcon]{DH}\frac{\log\big{(}\frac{1}{(1-\beta_{1})\log(q(|\gamma|+3))}\big{)}}{\log(q(|\gamma|+3))}$

derived from the zero repulsion phenomenon of Deuring and Heilbronn111Bombieri
[1, Ch. 6] proved that a bound on $N_{q}^{*}(\sigma,T)$ as in Theorem 2.1
suffices to prove (1.2) and (3.5). [3, (27)]. This contrasts with our use of
$N_{q}^{*}(\sigma,T)$ when $(\beta_{1},\chi_{1})$ exists and the
Vinogradov–Korobov zero-free region (2.1).

When $(\beta_{1},\chi_{1})$ does not exist and $q\geq\exp(\sqrt{\log x})$,
Theorem 1.1 achieves the asymptotic (3.4) in a stronger range than (3.3). When
$(\beta_{1},\chi_{1})$ exists, an exact comparison of (3.4) with Theorem 1.1
depends on how large $\beta_{1}$ is as an explicit function of $q$. We observe
here that we can estimate (2.2) using both the Vinogradov–Korobov zero-free
region and (3.5). This observation, when applied to Theorem 2.2, leads to a
mutual refinement of Theorem 1.1 and (3.4) in a range that improves upon
(3.3).

######  Corollary 3.3.

Let $q\geq 2$ and $a$ be coprime integers and $4\leq h\leq x$. If
$(\beta_{1},\chi_{1})$ exists and $\lambda$ is as in Theorem 1.1, then there
exist $\Cl[abcon]{Gal_{c}or},\Cl[abcon]{Gal_{c}or2}\in(0,1)$ such that if
$\lambda h/\varphi(q)\geq x^{\Cr{Gal_{c}or}}$, then

$\Big{|}1-\frac{\varphi(q)}{\lambda h}\sum_{\begin{subarray}{c}x-h<p\leq x\\\
p\equiv a\,(\mathrm{mod}{\,q})\end{subarray}}\log
p\Big{|}\ll(1-\beta_{1})(\log q)\exp\Big{(}\frac{-\Cr{Gal_{c}or2}\log x}{\log
q+(\log\frac{x}{h})^{\frac{2}{3}}(\log^{+}\log\frac{x}{h})^{\frac{1}{3}}+(\log
x)^{\frac{2}{5}}(\log\log x)^{\frac{1}{5}}}\Big{)}.$

The implied constant, $\Cr{Gal_{c}or}$, and $\Cr{Gal_{c}or2}$ are absolute and
effectively computable.

###### Proof.

Let $\theta$ be defined as in Theorem 1.1 and let
$\varepsilon\in(0,1-\theta)$. Without loss of generality, we may assume that
$\lambda h/\varphi(q)\geq x^{\theta+\varepsilon}$. The Deuring–Heilbronn zero-
free region (3.5) in Theorem 2.2 implies that

$\delta(t)\geq\frac{\Cr{DH}\log^{+}\big{(}\frac{1}{(1-\beta_{1})\log(qt)}\big{)}}{\log(qt)}\geq\frac{\Cr{DH}\log^{+}\big{(}\frac{1}{(1-\beta_{1})\log
x}\big{)}}{\log(qt)}$

for $e\leq t\leq\frac{x}{q}$. Consequently, if $e\leq t\leq\frac{x}{q}$, then

$\frac{x^{-\varepsilon^{2}\delta(t)}}{\sqrt{t}}\leq\sqrt{q}\exp\Big{(}-\frac{\varepsilon^{2}\Cr{DH}\log^{+}\big{(}\frac{1}{(1-\beta_{1})\log
x}\big{)}\log x}{\log(qt)}-\frac{1}{2}\log(qt)\Big{)}.$

The maximum value of this exponential over $t\in[\frac{ex}{h},\infty)$ occurs
at a value $t=t_{1}$ such that either $t_{1}=ex/h$, or
$\log(qt_{1})\asymp_{\varepsilon}(\log^{+}(\frac{1}{(1-\beta_{1})\log x})\log
x)^{1/2}$ and $t_{1}\geq ex/h$. Therefore, there exists an effectively
computable constant $\Cl[abcon]{DH-2}=\Cr{DH-2}({\varepsilon})>0$ such that if
$\lambda h/\varphi(q)\geq x^{\theta+\varepsilon}$, then

(3.6) $\displaystyle\sqrt{\frac{x}{h}}\frac{\log x}{\log
q}\sup\Big{\\{}\frac{x^{-\varepsilon^{2}\delta(t)}}{\sqrt{t}}\colon\frac{ex}{h}\leq
t\leq\frac{x^{(1-\theta)(1-\varepsilon)}}{q}\Big{\\}}+x^{-\varepsilon\theta/2}$
$\displaystyle\ll_{\varepsilon}\frac{\log x}{\log
q}\exp\Big{(}-\frac{\Cr{DH-2}\log^{+}(\tfrac{1}{(1-\beta_{1})\log x})\log
x}{\log\frac{qx}{h}+\big{(}\log^{+}(\tfrac{1}{(1-\beta_{1})\log x})\log
x\big{)}^{1/2}}\Big{)}+x^{-\varepsilon\theta/2}.$

This estimate, in conjunction with (2.2), leads to the best bound that Theorem
2.2 and (3.5) allow. We slightly weaken this bound for easy comparison with
(3.4). Let $0<\eta<1-\theta-\varepsilon$. Henceforth, assume that $\lambda
h/\varphi(q)\geq x^{1-\eta^{2}}$, in which case (1.7) implies

$\log
q\leq\log(qx/h)\leq(1+\varepsilon)\log(\varphi(q)x/h)+O_{\varepsilon}(1)\leq(1+\varepsilon)\eta^{2}\log
x+O_{\varepsilon}(1).$

Using $(1-\beta_{1})\log q\gg q^{-1/2}$ in (1.2), we obtain the bounds

$\log^{+}\Big{(}\frac{1}{(1-\beta_{1})\log
x}\Big{)}\leq\log^{+}\Big{(}\frac{1}{(1-\beta_{1})\log q}\Big{)}+O(1)\ll\log
q,\qquad x^{-1}\ll\big{(}(1-\beta_{1})\log q)^{1/\eta^{2}}.$

Collecting these estimates, we deduce that there exists effectively computable
constant $\Cr{DH-3}=\Cr{DH-3}(\varepsilon)$ depending at most on
$\varepsilon$, such that (3.6) is

$\ll_{\eta,\varepsilon}\frac{\log x}{\log q}\big{(}(1-\beta_{1})\log
x\big{)}^{\Cl[abcon]{DH-3}/\eta}=\big{(}\frac{\log x}{\log
q}\big{)}^{3}((1-\beta_{1})\log q)^{2}.$

The last equality follows upon fixing $\eta=\frac{\Cr{DH-3}}{2}$. Taking the
minimum of this estimate for (3.6) and the unconditional estimate (2.2), the
desired result follows from the basic inequality $\min\\{u,v\\}\leq
u^{1/2}v^{1/2}$ for $u,v>0$. Note that powers of $\frac{\log x}{\log q}$ are
absorbed into the exponential’s implied constant $\Cr{Gal_{c}or2}$. ∎

##  4\. A flexible prime number theorem: Proof of Theorem 2.2

We provide the details in the more involved case where $(\beta_{1},\chi_{1})$
exists using $N_{q}^{*}(\sigma,T)$ and $\theta=\frac{71}{75}$; the
complementary case is easier. All implied constants here are effectively
computable. We begin with a proof of (1.7).

######  Lemma 4.1.

If $(\beta_{1},\chi_{1})$ exists, $4\leq h\leq x$, and $x^{1/2}\leq h\leq x$,
then $\frac{1}{8}\min\\{1,(1-\beta_{1})\log x\\}<\lambda<2$.

###### Proof.

If $(\beta_{1},\chi_{1})$ exists, then per our definition,
$\frac{97}{100}<\beta_{1}<1$. If $\xi\in(x-h,x)$ satisfies (1.5), then we have
the explicit equality

$\xi=x\Big{(}\frac{\beta_{1}\frac{h}{x}}{1-(1-\frac{h}{x})^{\beta_{1}}}\Big{)}^{\frac{1}{1-\beta_{1}}}.$

The bounds $\frac{x}{3}<\xi<x$ follow from our ranges for $\beta$ and $h$.
Since $x\geq 4$, we have $\xi>1$ and $\log x\leq 5\log\xi$. We will use these
bounds repeatedly.

To bound $\lambda$ from above, we have

$\lambda=1-\chi_{1}(a)\xi^{\beta_{1}-1}<1+(x/3)^{\beta_{1}-1}<2.$

For lower bounds, we consider various cases. If $\chi_{1}(a)=-1$, then

$\min\\{1,(1-\beta_{1})\log x\\}\leq 1<1+\xi^{\beta_{1}-1}=\lambda.$

Recall that $\nu(U)=\min\\{1,(1-\beta_{1})\log U\\}$, so $0<\nu(\xi)\leq 1$.
If $\chi_{1}(a)=1$ and $\nu(\xi)=1$, then

$\frac{1}{8}\min\\{1,(1-\beta_{1})\log
x\\}\leq\frac{5}{8}\nu(\xi)=\frac{5}{8}<1-e^{-1}\leq
1-\xi^{\beta_{1}-1}=\lambda.$

If $\chi_{1}(a)=1$ and $0<\nu(\xi)<1$, then $\lambda=1-e^{-\nu(\xi)}$, and the
bound

$\frac{1}{8}\min\\{1,(1-\beta_{1})\log
x\\}\leq\frac{5}{8}\nu(\xi)<\frac{5}{8}\Big{(}1+\frac{\nu(\xi)}{2}+\frac{\nu(\xi)^{2}}{12}\Big{)}(1-e^{-\nu(\xi)})<\frac{95}{96}(1-e^{-\nu(\xi)})<\lambda$

follows from the Taylor expansion of $\frac{t}{1-e^{-t}}$ at $t=0$. These
cases exhaust all possibilities. ∎

###### Proof of Theorem 2.2.

Note that it suffices for us to consider $h\leq x-1$. Fix
$0<\varepsilon<1-\theta$, and assume that

(4.1) $\frac{\lambda h}{\varphi(q)}\geq x^{\theta+\varepsilon}.$

The explicit formula [2, Ch. 17–20] implies that if $x,T\geq 2$, then

$\sum_{k=1}^{\infty}\sum_{\begin{subarray}{c}p^{k}\leq x\\\ p\equiv
a\,(\mathrm{mod}{\,q})\end{subarray}}\log
p=\frac{x}{\varphi(q)}-\frac{\chi_{1}(a)}{\varphi(q)}\frac{x^{\beta_{1}}}{\beta_{1}}-\sum_{\chi}\frac{\bar{\chi}(a)}{\varphi(q)}\sideset{}{{}^{\prime}}{\sum}_{\begin{subarray}{c}\beta>0\\\
|\gamma|\leq
T\end{subarray}}\Big{(}\frac{x^{\rho}}{\rho}-\sum_{|\gamma|<1}\frac{1}{\rho}\Big{)}\\\
+O\Big{(}\frac{1}{\varphi(q)}\Big{(}\frac{x\log x+T}{T}\log q+\log
x+\frac{x(\log x)(\log T)}{T}\Big{)}+(\log x)(\log q)+\frac{x(\log
x)^{2}}{T}\Big{)},$

where $\sum_{\chi}$ is a sum over characters $\chi\,(\mathrm{mod}{\,q})$ and
$\sum^{\prime}$ is a sum over non-trivial zeros
$\rho=\beta+i\gamma\neq\beta_{1}$ of $L(s,\chi)$. The contribution from
$p^{k}\leq x$ with $k\geq 2$ is trivially bounded by $\ll\sqrt{x}$ using
Chebyshev’s bound $\sum_{p\leq x}\log p\ll x$, leading to the equation

(4.2) $\sum_{\begin{subarray}{c}p\leq x\\\ p\equiv
a\,(\mathrm{mod}{\,q})\end{subarray}}\log
p=\frac{x}{\varphi(q)}-\frac{\chi_{1}(a)}{\varphi(q)}\frac{x^{\beta_{1}}}{\beta_{1}}-\sum_{\chi}\frac{\bar{\chi}(a)}{\varphi(q)}\sideset{}{{}^{\prime}}{\sum}_{\begin{subarray}{c}\beta>0\\\
|\gamma|\leq
T\end{subarray}}\Big{(}\frac{x^{\rho}}{\rho}-\sum_{|\gamma|<1}\frac{1}{\rho}\Big{)}\\\
+O\Big{(}\frac{1}{\varphi(q)}\Big{(}\frac{x\log x+T}{T}\log q+\log
x+\frac{x(\log x)(\log T)}{T}\Big{)}+(\log x)(\log q)+\frac{x(\log
x)^{2}}{T}+\sqrt{x}\Big{)}.$

We apply (4.2) twice with $T=x^{(1-\theta)(1-\varepsilon)}$ (once with $p\leq
x$ and once with $p\leq x-h$). Our assumption (4.1) then yields

$\displaystyle\mathcal{E}:=\Big{|}\frac{\lambda
h}{\varphi(q)}-\sum_{\begin{subarray}{c}x-h<p\leq x\\\ p\equiv
a\,(\mathrm{mod}{\,q})\end{subarray}}\log
p\Big{|}\ll_{\varepsilon}\frac{1}{\varphi(q)}\sum_{\chi}~{}\sideset{}{{}^{\prime}}{\sum}_{\begin{subarray}{c}\beta>0\\\
|\gamma|\leq
x^{(1-\theta)(1-\varepsilon)}\end{subarray}}\Big{|}\frac{x^{\rho}-(x-h)^{\rho}}{\rho}\Big{|}+\frac{\lambda
h}{\varphi(q)}x^{-\varepsilon\theta/2}.$

Recall that $h\leq x-1$. Using the bounds

$\displaystyle\Big{|}\frac{x^{\rho}-(x-h)^{\rho}}{\rho}\Big{|}$
$\displaystyle=\Big{|}\int_{x-h}^{x}\tau^{\rho-1}d\tau\Big{|}\ll\int_{x-h}^{x}\tau^{\beta-1}d\tau\ll
h(x-h)^{\beta-1},$
$\displaystyle\Big{|}\frac{x^{\rho}-(x-h)^{\rho}}{\rho}\Big{|}$
$\displaystyle\ll\frac{x^{\beta}}{|\rho|}$

as well as Taylor’s theorem for $\frac{x^{s}-(x-h)^{s}}{s}$ when
$|s|\leq\frac{1}{\log x}$, we find that

$\Big{|}\frac{x^{\rho}-(x-h)^{\rho}}{\rho}\Big{|}\ll\begin{cases}hx^{\beta-1}\log
x&\mbox{if $0<\beta<\frac{1}{4}$ and $|\gamma|\leq ex/h$,}\\\
hx^{\beta-1}&\mbox{if $\beta\geq\frac{1}{4}$ and $|\gamma|\leq ex/h$,}\\\
x^{\beta}/|\gamma|&\mbox{if $|\gamma|>ex/h$.}\end{cases}$

(Separately consider the cases where $h\leq\frac{x}{2}$ and $\frac{x}{2}<h\leq
x-1$.) For $t\in\mathbb{R}$, there are $\ll\log(q(|t|+1))$ zeros of
$L(s,\chi)$ such that $|t-\gamma|\leq 1$ [2, §16]. Once we use this to
trivially estimate the contribution from the $\rho$ with
$\beta\leq\theta+\varepsilon-\varepsilon\theta$, it follows from (4.1) that

(4.3)
$\mathcal{E}\ll_{\varepsilon}\frac{1}{\varphi(q)}\sum_{\chi}~{}\sideset{}{{}^{\prime}}{\sum}_{\begin{subarray}{c}\beta>\theta+\varepsilon-\varepsilon\theta\\\
|\gamma|\leq
ex/h\end{subarray}}\frac{x^{\beta}}{x/h}+\frac{1}{\varphi(q)}\sum_{\chi}~{}\sideset{}{{}^{\prime}}{\sum}_{\begin{subarray}{c}\beta>\theta+\varepsilon-\varepsilon\theta\\\
ex/h<|\gamma|\leq
x^{(1-\theta)(1-\varepsilon)}\end{subarray}}\frac{x^{\beta}}{|\gamma|}+\frac{\lambda
h}{\varphi(q)}x^{-\varepsilon\theta/2}.$

We dyadically decompose $[ex/h,x^{(1-\theta)(1-\varepsilon)}]$ into $O(\log
x)$ disjoint subintervals, so that (4.3) leads to

(4.4)
$\mathcal{E}\ll_{\varepsilon}\frac{1}{\varphi(q)}\sum_{k=\lceil\log(ex/h)\rceil}^{\lceil(1-\theta)(1-\varepsilon)\log
x\rceil}\Big{(}\sum_{\chi}~{}\sideset{}{{}^{\prime}}{\sum}_{\begin{subarray}{c}\beta>\theta+\varepsilon-\varepsilon\theta\\\
|\gamma|\leq e^{k}\end{subarray}}\frac{x^{\beta}}{e^{k}}\Big{)}+\frac{\lambda
h}{\varphi(q)}x^{-\varepsilon\theta/2}.$

We claim that to prove Theorem 1.1, it suffices to show that if $ex/h\leq
T\leq ex^{(1-\theta)(1-\varepsilon)}$, then

(4.5)
$\sum_{\chi}~{}\sideset{}{{}^{\prime}}{\sum}_{\begin{subarray}{c}\beta>\theta+\varepsilon-\varepsilon\theta\\\
|\gamma|\leq
T\end{subarray}}\frac{x^{\beta}}{T}\ll_{\varepsilon}\frac{\lambda\log x}{\log
q}\Big{(}\frac{x}{\sqrt{T}}\sup\Big{\\{}\frac{x^{-\varepsilon^{2}\delta(t)}}{\sqrt{t}}\colon\frac{ex}{h}\leq
t\leq\frac{x^{(1-\theta)(1-\varepsilon)}}{q}\Big{\\}}+qx^{\theta+\varepsilon-\varepsilon\theta}\log
x\Big{)}.$

Assuming the claim, we apply (4.4) and (4.5), summing over $k$ to obtain

$\mathcal{E}\ll_{\varepsilon}\frac{\lambda
h}{\varphi(q)}\Big{(}\sqrt{\frac{x}{h}}\frac{\log x}{\log
q}\sup\Big{\\{}\frac{x^{-\varepsilon^{2}\delta(t)}}{\sqrt{t}}\colon\frac{ex}{h}\leq
t\leq\frac{x^{(1-\theta)(1-\varepsilon)}}{q}\Big{\\}}+\frac{qx^{\theta+\varepsilon-\varepsilon\theta}\log
x}{h}+x^{-\varepsilon\theta/2}\Big{)}.$

Notice that $x^{\theta+\varepsilon}\leq\frac{\lambda
h}{\varphi(q)}<\frac{2h}{\varphi(q)}$ by Lemma 4.1 and (4.1). Also, since
$h\leq x$, it follows that $\log q\ll_{\varepsilon}\log x$, hence
$q\ll_{\varepsilon}\varphi(q)x^{\varepsilon\theta/4}$. Overall, this shows
that $qx^{\theta+\varepsilon-\varepsilon\theta}\log
x\ll_{\varepsilon}hx^{-\varepsilon\theta/2}$. This establishes Theorem 2.2
with the above estimate for $\mathcal{E}$; it remains to prove the claim.

Let $ex/h\leq T\leq ex^{(1-\theta)(1-\varepsilon)}$. By partial summation and
Theorem 2.1, we find that

$\displaystyle\sum_{\chi}~{}\sideset{}{{}^{\prime}}{\sum}_{\begin{subarray}{c}\beta>\theta+\varepsilon-\varepsilon\theta\\\
|\gamma|\leq T\end{subarray}}\frac{x^{\beta}}{T}$
$\displaystyle\leq\frac{x^{\theta+\varepsilon-\varepsilon\theta}}{T}N_{q}^{*}\Big{(}\theta+\varepsilon-\varepsilon\theta,T\Big{)}+\frac{x\log
x}{T}\int_{\theta+\varepsilon-\varepsilon\theta}^{1-\delta(T)}N_{q}^{*}(\sigma,T)x^{\sigma-1}\mathrm{d}\sigma$
$\displaystyle\ll_{\varepsilon}\nu(qT)\Big{(}\frac{x^{\theta+\varepsilon-\varepsilon\theta}}{T}(qT)^{(\frac{1+\varepsilon}{1-\theta})(1-\theta-\varepsilon+\varepsilon\theta)}+\frac{x\log
x}{T}\int_{\theta+\varepsilon-\varepsilon\theta}^{1-\delta(T)}\Big{(}\frac{(qT)^{\frac{1+\varepsilon}{1-\theta}}}{x}\Big{)}^{1-\sigma}\mathrm{d}\sigma\Big{)}$
$\displaystyle\ll_{\varepsilon}\nu(qT)\Big{(}q^{1-\varepsilon^{2}}x^{\theta+\varepsilon-\varepsilon\theta}\log
x+\frac{x\log
x}{|\log(x/(qT)^{\frac{1+\varepsilon}{1-\theta}})|+1}\cdot\frac{1}{T}\Big{(}\frac{(qT)^{\frac{1+\varepsilon}{1-\theta}}}{x}\Big{)}^{\delta(T)}\Big{)}.$

Since $T\leq ex^{(1-\theta)(1-\varepsilon)}$ and $\log q\ll_{\varepsilon}\log
x$, the definition of $\nu$ implies that
$\nu(qT)\ll_{\varepsilon}\nu(q)\frac{\log x}{\log q}$. The bounds for
$\lambda$ in (1.7) and the inequality $q^{1-\varepsilon^{2}}\leq q$ then imply
that

(4.6)
$\sum_{\chi}~{}\sideset{}{{}^{\prime}}{\sum}_{\begin{subarray}{c}\beta>\theta+\varepsilon-\varepsilon\theta\\\
|\gamma|\leq
T\end{subarray}}\frac{x^{\beta}}{T}\ll_{\varepsilon}\frac{\lambda\log x}{\log
q}\Big{(}qx^{\theta+\varepsilon-\varepsilon\theta}\log x+\frac{x\log
x}{|\log(x/(qT)^{\frac{1+\varepsilon}{1-\theta}})|+1}\cdot\frac{1}{T}\Big{(}\frac{(qT)^{\frac{1+\varepsilon}{1-\theta}}}{x}\Big{)}^{\delta(T)}\Big{)}.$

To estimate the second term in the parentheses, consider cases depending on
the size of $T$. First, suppose that $x^{(1-\theta)(1-\varepsilon)}/q\leq
T\leq ex^{(1-\theta)(1-\varepsilon)}$. This implies that the ratio
$(qT)^{\frac{1+\varepsilon}{1-\theta}}/x$ may exceed $1$. Since
$0\leq\delta(T)\leq 1-\theta-\varepsilon<(1-\theta)(1-\varepsilon)$, this
implies that

$\displaystyle\frac{x\log
x}{|\log(x/(qT)^{\frac{1+\varepsilon}{1-\theta}})|+1}\cdot\frac{1}{T}\Big{(}\frac{(qT)^{\frac{1+\varepsilon}{1-\theta}}}{x}\Big{)}^{\delta(T)}$
$\displaystyle\ll\frac{x\log
x}{T}\min\Big{[}1,\Big{(}\frac{(qT)^{\frac{1+\varepsilon}{1-\theta}}}{x}\Big{)}^{(1-\theta)(1-\varepsilon)}\Big{]}$
$\displaystyle\ll_{\varepsilon}\min\Big{[}\frac{x\log
x}{T},q^{1-\varepsilon^{2}}x^{\theta+\varepsilon-\varepsilon\theta}T^{-\varepsilon^{2}}\Big{]}$
$\displaystyle\ll_{\varepsilon}qx^{\theta+\varepsilon-\varepsilon\theta}\log
x,$

because $T\geq x^{(1-\theta)(1-\varepsilon)}/q$ and $T\geq e$ by assumption.
This proves the claim (4.5) when $x^{(1-\theta)(1-\varepsilon)}/q\leq T\leq
ex^{(1-\theta)(1-\varepsilon)}$. Finally, suppose $ex/h\leq
T<x^{(1-\theta)(1-\varepsilon)}/q$ so

$\displaystyle\frac{x\log
x}{|\log(x/(qT)^{\frac{1+\varepsilon}{1-\theta}})|+1}\cdot\frac{1}{T}\Big{(}\frac{(qT)^{\frac{1+\varepsilon}{1-\theta}}}{x}\Big{)}^{\delta(T)}$
$\displaystyle\ll_{\varepsilon}\frac{x^{1-\varepsilon^{2}\delta(T)}}{T}$
$\displaystyle\ll_{\varepsilon}\frac{x}{\sqrt{T}}\sup\Big{\\{}\frac{x^{-\varepsilon^{2}\delta(t)}}{\sqrt{t}}\colon\frac{ex}{h}\leq
t\leq\frac{x^{(1-\theta)(1-\varepsilon)}}{q}\Big{\\}}.$

These observations and (4.6) complete the proof of the claim (4.5) and hence
Theorem 2.2. ∎

Remark 1.2 follows by using (1.6) instead of (1.7) to deduce (4.6). Remark 1.3
follows by using the bound for $N_{q}(\sigma,T)$ in Theorem 2.1 instead of
$N_{q}^{*}(\sigma,T)$ in our proof of Theorem 2.2; this follows because
$\nu(qT)\asymp 1$ when $\beta_{1}=1-\frac{\Cr{relative}}{\log q}$ for some
fixed constant $\Cr{relative}$.

##  5\. Proof of Theorem 2.1

The claimed bound on $N_{q}(\sigma,T)$ follows from the combined work of
Huxley [6] and Jutila [8]; see [4]. We refine Bombieri’s modifications [1, Ch.
6] to Gallagher’s work [3] to prove the claimed bound for
$N_{q}^{*}(\sigma,T)$. Throughout the proof, we assume that
$(\beta_{1},\chi_{1})$ exists. All implied constants in this proof are
effectively computable. Since $L(s,\chi)$ has no real nontrivial zero when the
modulus of $\chi$ is at most $2\times 10^{6}$ [19], we may assume that $q\geq
2\times 10^{6}$.

###  5.1. Preliminaries

Let $\chi\,(\mathrm{mod}{\,q})$ be a primitive Dirichlet character, let
$\chi_{2}$ be the primitive character induced by $\chi\chi_{1}$, and let
$\delta(\cdot)$ be the indicator function for the trivial character. Define
the constants

(5.1) $\alpha=26.354133491747653,\quad A=1.239475274727766,\quad B=10^{-16}.$

Let $\tau\in\mathbb{R}$ and

(5.2) $T\geq 100,\quad|\tau|\leq T,\quad\text{$|\tau|\geq 10$ if
$\delta(\chi)=1$},\quad\frac{1}{\log(qT)}<\eta<\frac{1}{2A},\quad
s_{0}=1+B\eta+i\tau.$

######  Lemma 5.1 ([12], p. 331).

If $t\in\mathbb{R}$ and $r>1/\log(q(|t|+3))$, then the number
$n_{\chi}(r,1+it)$ of zeros $\rho$ of $L(s,\chi)$ such that $|1+it-\rho|\leq
r$ is $\ll r\log(q(|t|+1))$.

Let $k\geq 2$ be an integer. From the Euler product and Hadamard factorization
for $L(s,\chi)$, we find that if $\mathrm{Re}(s)>1$, then

(5.3)
$\frac{(-1)^{k+1}}{k!}\Big{(}\frac{L^{\prime}}{L}\Big{)}^{(k)}(s,\chi)=\sum_{n=1}^{\infty}\frac{\chi(n)\Lambda(n)}{n^{s}}\frac{(\log
n)^{k}}{k!}=\frac{\delta(\chi)}{(s-1)^{k+1}}-\sum_{L(\rho,\chi)=0}\frac{1}{(s-\rho)^{k+1}}+O(1),$

where the sum is over all nontrivial zeros $\rho=\beta+i\gamma$ of $L(s,\chi)$
with $\beta>0$ and $\gamma\in\mathbb{R}$. Write $s=\sigma+it$ with $\sigma>1$
and $t\in\mathbb{R}$, and define

$F(s)=\Big{(}\frac{L^{\prime}}{L}\Big{)}^{(k)}(s,\chi)+\Big{(}\frac{L^{\prime}}{L}\Big{)}^{(k)}(s+1-\beta_{1},\chi_{2}).$

###  5.2. A lower bound for $|F(s_{0})|$

It follows from (5.3) that if $\mathrm{Re}(s)>1$, then

$\frac{(-1)^{k}}{k!}F(s_{0})+\frac{\delta(\chi)}{(s_{0}-1)^{k+1}}+\frac{\delta(\chi_{2})}{(s_{0}-\beta_{1})^{k+1}}\\\
=\sum_{L(\rho,\chi)=0}\frac{1}{(s_{0}-\rho)^{k+1}}+\sum_{L(\rho^{\prime},\chi_{2})=0}\frac{1}{(s_{0}+1-\beta_{1}-\rho^{\prime})^{k+1}}+O(1).$

Since $(\beta_{1},\chi_{1})$ exists, $\beta_{1}$ is a simple zero of
$\prod_{\chi\,(\mathrm{mod}{\,q})}L(s,\chi)$ with $L(\beta_{1},\chi_{1})=0$,
so

$\frac{(-1)^{k}}{k!}F(s_{0})+\delta(\chi)\Big{(}\frac{1}{(s_{0}-1)^{k+1}}-\frac{1}{(s_{0}+1-2\beta_{1})^{k+1}}\Big{)}\\\
=\sum_{\begin{subarray}{c}L(\rho,\chi)=0\\\
\rho\neq\beta_{1}\end{subarray}}\frac{1}{(s_{0}-\rho)^{k+1}}+\sum_{\begin{subarray}{c}L(\rho^{\prime},\chi_{2})=0\\\
\rho^{\prime}\neq\beta_{1}\end{subarray}}\frac{1}{(s_{0}+1-\beta_{1}-\rho^{\prime})^{k+1}}+O(1).$

The constraints in (5.2) ensure that

(5.4) $\frac{(-1)^{k}}{k!}F(s_{0})=\sum_{\begin{subarray}{c}L(\rho,\chi)=0\\\
\rho\neq\beta_{1}\end{subarray}}\frac{1}{(s_{0}-\rho)^{k+1}}+\sum_{\begin{subarray}{c}L(\rho^{\prime},\chi_{2})=0\\\
\rho^{\prime}\neq\beta_{1}\end{subarray}}\frac{1}{(s_{0}+1-\beta_{1}-\rho^{\prime})^{k+1}}+O(1).$

If $|1+i\tau-\rho|>A\eta$, then $|s_{0}-\rho|\geq\eta\sqrt{A^{2}+B^{2}}$.
Therefore, by Lemma 5.1, we have

(5.5)
$\sum_{|1+i\tau-\rho|>A\eta}\frac{1}{|s_{0}-\rho|^{k+1}}\leq\frac{\eta\sqrt{A^{2}+B^{2}}}{(\eta\sqrt{A^{2}+B^{2}})^{k}}\sum_{\rho}\frac{1}{|s_{0}-\rho|^{2}}\ll\frac{\log(qT)}{(\eta\sqrt{A^{2}+B^{2}})^{k}}.$

By (5.4) and (5.5), we have that

(5.6) $\displaystyle\frac{(-1)^{k}}{k!}F(s_{0})$
$\displaystyle=\sum_{\begin{subarray}{c}L(\rho,\chi)=0\\\ \rho\neq\beta_{1}\\\
|1+i\tau-\rho|\leq A\eta\end{subarray}}\frac{1}{(s_{0}-\rho)^{k+1}}$
$\displaystyle+\sum_{\begin{subarray}{c}L(\rho^{\prime},\chi_{2})=0\\\
\rho^{\prime}\neq\beta_{1}\\\ |1+i\tau-\rho^{\prime}|\leq
A\eta\end{subarray}}\frac{1}{(s_{0}+1-\beta_{1}-\rho^{\prime})^{k+1}}+O\Big{(}\frac{\log(qT)}{(\eta\sqrt{A^{2}+B^{2}})^{k}}\Big{)}.$

In order to obtain a lower bound for (5.4), we use the following result of
Kolesnik and Straus [9] which numerically refines the lower bound on power
sums proved by Sós and Turán [20].

######  Theorem 5.2.

Let $M\geq 0$ and $N\geq 1$ be integers, and let
$z_{1},\ldots,z_{N}\in\mathbb{C}$. There exists $k\in[M+1,M+N]\cap\mathbb{Z}$
such that $|z_{1}^{k}+\cdots+z_{N}^{k}|\geq
1.007(\frac{N}{4e(M+N)})^{N}|z_{1}|^{k}$.

######  Remark 5.3.

Makai [13] proved that the constant $4e$ in the denominator is sharp.

In order to apply Theorem 5.2, we proceed under

######  Hypothesis 5.4 (H$(\tau,\eta)$).

There exists a zero $\rho_{0}$ of $L(s,\chi)$ such that
$|1+i\tau-\rho_{0}|\leq\eta$.

Assuming H$(\tau,\eta)$, we have
$N:=n_{\chi}(A\eta;1+i\tau)+n_{\chi_{2}}(A\eta,1+i\tau)\geq 1$. Let $M\geq 0$.
By Theorem 5.2, H$(\tau,\eta)$ implies that there exists $k\in[M,M+N-1]$ such
that

(5.7) $\Big{|}\sum_{\begin{subarray}{c}L(\rho,\chi)=0\\\ \rho\neq\beta_{1}\\\
|1+it-\rho|\leq
A\eta\end{subarray}}\frac{1}{(s_{0}-\rho)^{k+1}}+\sum_{\begin{subarray}{c}L(\rho^{\prime},\chi_{2})=0\\\
\rho^{\prime}\neq\beta_{1}\\\ |1+it-\rho^{\prime}|\leq
A\eta\end{subarray}}\frac{1}{(s_{0}+1-\beta_{1}-\rho^{\prime})^{k+1}}\Big{|}\\\
\geq
1.007\Big{(}\frac{N}{4e(M+N)}\Big{)}^{N}\frac{1}{|s_{0}-\rho_{0}|^{k+1}}\geq\Big{(}\frac{1}{4e(1+\frac{M}{N})}\Big{)}^{N}\frac{1}{(\eta(B+1))^{k+1}}.$

The following lemma provides a strong upper bound for $N$ that depends on the
best permissible value of $\phi>0$ such that for all $q\geq 2$ and all
$t\in\mathbb{R}$, we have the bound

(5.8) $|L(\tfrac{1}{2}+it,\chi)|\ll(q(|t|+1))^{\phi}.$

######  Lemma 5.5.

Let $\chi$ be a Dirichlet character modulo $q\geq 2$, and let $\phi$ be as in
(5.8). If $\tau\in\mathbb{R}$, $T\geq 100$, $|t|\leq T$, and
$r\in(\frac{1}{\log(qT)},\frac{1}{2})$, then $n_{\chi}(r;1+it)\leq 4\phi
r\log(qT)+O(1)$.

###### Proof.

If $\delta(\chi)=1$, then $n_{\chi}(\frac{1}{2},1+i\tau)=0$ for all
$|\tau|\leq 13.5$. Therefore, we may assume that $|\tau|>13.5$ when
$\delta(\chi)=1$. Choose $R\in(2r,\frac{1}{2}+r)$ so that
$\\{w\in\mathbb{C}\colon|1+r+it-w|=R\\}$ contains no zeros of $L(z,\chi)$. By
[4, Lemma 3.2] with $f(z)=L(z,\chi)$ and $a=s:=1+r+it$, we have

(5.9)
$\mathrm{Re}\Big{[}\frac{L^{\prime}}{L}(s,\chi)\Big{]}=\sum_{|s-\rho|<R}\mathrm{Re}\Big{[}\frac{1}{s-\rho}-\frac{s-\rho}{R^{2}}\Big{]}+J,\quad
J:=\int_{0}^{2\pi}\frac{\cos\theta}{\pi
R}\log|L(s+Re^{i\theta},\chi)|d\theta.$

We will bound $J$ from below; to this end, we write
$J=\int_{0}^{\pi/2}+\int_{\pi/2}^{3\pi/2}+\int_{3\pi/2}^{2\pi}=J_{1}+J_{2}+J_{3}$,
say, so we may consider each contribution separately. For $J_{1}$, note that

$\log|L(s+Re^{i\theta},\chi)|\leq\log\zeta(1+r+R\cos\theta)\ll\log\Big{(}\frac{1}{r+R\cos\theta}\Big{)}.$

Writing $I_{1}=[0,\tfrac{\pi}{2}-r]$ and
$I_{2}=[\tfrac{\pi}{2}-r,\tfrac{\pi}{2}]$, we have that

$J_{1}=\int_{I_{1}}+\int_{I_{2}}\ll\int_{I_{1}}\cos\theta\log(1/\cos\theta)d\theta+\log(1/r)\int_{I_{2}}\cos\theta
d\theta\ll 1+r\log(1/r)\ll 1.$

A similar argument holds for $J_{3}$ so $J_{1}+J_{3}\ll 1$. For $J_{2}$,
consider $\theta\in[\frac{\pi}{2},\frac{3\pi}{2}]$. Since
$R\in(2r,\frac{1}{2}+r)$, we have that $\frac{1}{2}<1+r+R\cos\theta\leq 1+r$.
Write $\sigma^{\prime}=1+r+R\cos\theta$ and $y=t+R\sin\theta$ so that
$s+Re^{i\theta}=\sigma^{\prime}+iy$. Let $\delta(\chi)=1$ if $\chi$ is trivial
and $\delta(\chi)=0$ otherwise. It now follows from (5.8), the bound
$|L(1+r+iu,\chi)|\ll r^{-1}$ for all $u\in\mathbb{R}$, and the
Phragmén–Lindelöf principle that

$\Big{|}\frac{\sigma^{\prime}+iy-1}{\sigma^{\prime}+iy+1}\Big{|}^{\delta(\chi)}|L(\sigma^{\prime}+iy,\chi)|\ll((q(|y|+1))^{\phi})^{\frac{1+r-\sigma^{\prime}}{\frac{1}{2}+r}}(r^{-1})^{1-\frac{1+r-\sigma^{\prime}}{\frac{1}{2}+r}}\ll\frac{(q(|t|+1))^{\frac{2R\phi}{2r+1}\cos\theta}}{r^{1+\frac{2R}{2r+1}\cos\theta}}.$

Per the discussion at the beginning of the proof, if $\delta(\chi)=1$, then
$|y|\geq 12$. Therefore, regardless of what $\delta(\chi)$ equals, it follows
that

$\log|L(s+Re^{i\theta},\chi)|\leq\frac{2R\phi\cos\theta}{2r+1}\log(q(|t|+1))+O(\log
r^{-1}).$

This yields

$J_{2}\geq-\frac{2\phi\log(q(|t|+1))}{(2r+1)\pi}\int_{\pi/2}^{3\pi/2}(\cos\theta)^{2}d\theta+O(\log
r^{-1})\quad\textup{and}\quad J\geq-\frac{\phi\log(q(|t|+1))}{2r+1}+O(\log
r^{-1}).$

For the sum over zeros in (5.9), we observe that the terms are non-negative.
Therefore, we may restrict the sum over zeros $\rho$ with $|s-\rho|<R$ to the
concentric circle $|s-\rho|\leq 2r$ and deduce that

$\sum_{|s-\rho|<R}\mathrm{Re}\Big{[}\frac{1}{s-\rho}-\frac{s-\rho}{R^{2}}\Big{]}\geq\sum_{|s-\rho|\leq
2r}\mathrm{Re}\Big{[}\frac{1}{s-\rho}-\frac{s-\rho}{R^{2}}\Big{]}\geq
r\Big{(}\frac{1}{4r^{2}}-\frac{1}{R^{2}}\Big{)}n_{\chi}(2r;1+r+it).$

We conclude that

$n_{\chi}(r;1+it)\leq
n_{\chi}(2r;1+r+it)\leq\frac{4rR^{2}}{R^{2}-r^{2}}\Big{(}\frac{\phi\log(q(|t|+3))}{2r+1}+\mathrm{Re}\Big{[}\frac{L^{\prime}}{L}(s,\chi)\Big{]}+O(\log
r^{-1})\Big{)}.$

We use the bound
$\mathrm{Re}(\frac{L^{\prime}}{L}(s,\chi))\leq-\frac{\zeta^{\prime}}{\zeta}(1+r)\leq\frac{1}{r}$
to obtain the desired result with
$R=\frac{1}{4}+\frac{r}{2}+\frac{1}{2}(\frac{r}{2}+r^{2})^{1/2}$ when
$r<\frac{1}{6}$ and $R=\frac{1}{4}+\frac{3r}{2}$ otherwise. ∎

By Lemma 5.5 and the fact that $\chi_{2}$ is a Dirichlet character modulo $q$,
we find that there exists an absolute constant $K_{0}\geq 0$ such that

(5.10)
$N\leq\mathcal{N}=\mathcal{N}(\eta,K,q,T):=8A\phi\eta\log(qT)+K\qquad\textup{for
all $K\geq K_{0}$.}$

Let $\alpha$ be as in (5.1), and define

(5.11) $M:=(\alpha-1)\mathcal{N}.$

By (5.6) and (5.7), we find that

$\Big{|}\frac{\eta^{k+1}}{k!}F(s_{0})\Big{|}\geq
1.007\Big{(}\frac{1}{4e\alpha}\Big{)}^{\mathcal{N}}\frac{1}{(B+1)^{k+1}}-O\Big{(}\frac{\eta\log(qT)}{(\sqrt{A^{2}+B^{2}})^{k}}\Big{)}\\\
=1.007\Big{(}\frac{1}{4e\alpha}\Big{)}^{\mathcal{N}}\frac{1}{(B+1)^{k+1}}\Big{(}1-O\Big{(}\Big{(}\frac{B+1}{\sqrt{A^{2}+B^{2}}}\Big{)}^{k+1}(4e\alpha)^{\mathcal{N}}\eta\log(qT)\Big{)}\Big{)}.$

Since $(\alpha-1)\mathcal{N}+1\leq k+1\leq\alpha\mathcal{N}$ and
$B+1<\sqrt{A^{2}+B^{2}}$ by (5.1), it follows that

$\Big{|}\frac{\eta^{k+1}}{k!}F(s_{0})\Big{|}\geq
1.007\Big{(}\frac{1}{4e\alpha(B+1)^{\alpha}}\Big{)}^{\mathcal{N}}\Big{(}1-O\Big{(}\Big{(}4e\alpha\Big{(}\frac{B+1}{\sqrt{A^{2}+B^{2}}}\Big{)}^{\alpha}\Big{)}^{\mathcal{N}}\eta\log(qT)\Big{)}\Big{)}\\\
=1.007\Big{(}\Big{(}\frac{1}{4e\alpha(B+1)^{\alpha}}\Big{)}^{A}\Big{)}^{8\phi\eta\log(qT)+\frac{K}{A}}\Big{(}1-O\Big{(}\Big{(}4e\alpha\Big{(}\frac{B+1}{\sqrt{A^{2}+B^{2}}}\Big{)}^{\alpha}\Big{)}^{8A\phi\eta\log(qT)+K}\eta\log(qT)\Big{)}\Big{)}.$

We chose the values of $A$, $B$, and $\alpha$ in (5.1) to approximate the
infimum222The infimum $1110.817286401673\ldots$ is achieved when
$\alpha=\alpha_{0}=26.354133491747653\ldots$, $A=(4e\alpha)^{1/\alpha}$, and
$B=0$. The number $\alpha_{0}$ solves the equation
$\alpha_{0}-(\log\alpha_{0})^{2}-(1+\log 16)\log\alpha_{0}=\log 4+(\log
4)^{2}$. of $(4e\alpha(B+1)^{\alpha})^{A}$ subject to the constraints $A>1$,
$B>0$, $\alpha>1$, and $4e\alpha(\frac{B+1}{\sqrt{A^{2}+B^{2}}})^{\alpha}<1$,
leading us to

$(4e\alpha(B+1)^{\alpha})^{A}=1110.817286401682\ldots,\qquad
0<4e\alpha\Big{(}\frac{B+1}{\sqrt{A^{2}+B^{2}}}\Big{)}^{\alpha}<1-10^{-14}.$

Once we choose $K$ in (5.10) to be a sufficiently large multiple of
$1+\frac{1}{\phi}$, we conclude that

$\Big{|}\frac{\eta^{k+1}}{k!}F(s_{0})\Big{|}\geq\Big{(}\frac{1}{1110.817286401682\ldots}\Big{)}^{8\phi\eta\log(qT)+\frac{K}{A}}\geq\frac{1}{(1.250015191)^{M}}.$

###  5.3. An upper bound for $|F(s_{0})|$

Define $j_{k}(u)=e^{-Bu}u^{k}/k!$, and consider the identity

$\Big{|}\frac{\eta^{k+1}}{k!}F(s_{0})\Big{|}=\eta\Big{|}\sum_{n=1}^{\infty}\frac{\Lambda(n)(\chi(n)+\chi_{2}(n)n^{\beta_{1}-1})}{n^{1+i\tau}}j_{k}(\eta\log
n)\Big{|}.$

Let $A_{0}=0.2919\ldots$ and $A_{1}=1.729\ldots\times 10^{18}$ be solutions
for the equations $A_{0}e=(1.26)^{-1}=A_{1}e^{1-A_{1}B/4}$. By (5.1), (5.10),
and (5.11), we have that
$k\in[(\alpha-1)\mathcal{N},\alpha\mathcal{N}]\subseteq[M,\frac{\alpha}{\alpha-1}M]$.
Define

$N_{0}:=\exp(A_{0}M/\eta),\qquad N_{1}:=\exp(A_{1}M/\eta).$

If $n\leq N_{0}$, then $\eta\log n\leq A_{0}M$. Since $k\in[M,2M]$ and
$k!\geq(k/e)^{k}$, it follows that

$j_{k}(\eta\log n)=n^{-B\eta}\frac{(\eta\log n)^{k}}{k!}\leq
n^{-B\eta}\Big{(}\frac{e\eta\log n}{k}\Big{)}^{k}\leq
n^{-B\eta}\Big{(}\frac{A_{0}eM}{k}\Big{)}^{k}\leq n^{-B\eta}(1.26)^{-k}.$

If $n\geq N_{1}$, then $\eta\log n\geq A_{1}M$. Since $e^{-Bu/2}u^{k}/k!$ is
decreasing in the range $u>2k/B$, we find that if $n\geq N_{1}$, then

$j_{k}(\eta\log n)=n^{-\frac{B\eta}{2}}\frac{e^{-\frac{B}{2}\eta\log
n}(\eta\log n)^{k}}{k!}\leq
n^{-\frac{B\eta}{2}}\frac{e^{-\frac{B}{4}A_{1}M}(A_{1}M)^{k}}{k!}\leq
n^{-\frac{B\eta}{2}}(1.26)^{-k}.$

It follows that

$\eta\Big{|}\sum_{n\notin[N_{0},N_{1}]}\frac{\Lambda(n)(\chi(n)+\chi_{2}(n)n^{\beta_{1}-1})}{n^{1+i\tau}}j_{k}(\eta\log
n)\Big{|}\leq\frac{2\eta}{(1.26)^{k}}\sum_{n=1}^{\infty}\frac{\Lambda(n)}{n^{1+\frac{B\eta}{2}}}\ll\frac{1}{(1.26)^{k}}.$

Since $|\chi(n)+\chi_{2}(n)n^{\beta_{1}-1}|\leq 2$, $0\leq j_{k}(\eta\log
n)\leq 1$, and the number of composite prime powers up to $X\geq 1$ is
$\ll\sqrt{X}/\log X$, it follows from our ranges of $\eta$ and $k$ that

$\displaystyle\eta\Big{|}\sum_{\begin{subarray}{c}n\in[N_{0},N_{1}]\\\
\textup{$n$
composite}\end{subarray}}\frac{\Lambda(n)(\chi(n)+\chi_{2}(n)n^{\beta_{1}-1})}{n^{1+i\tau}}j_{k}(\eta\log
n)\Big{|}\ll\sum_{\begin{subarray}{c}n\in[N_{0},N_{1}]\\\ \textup{$n$
composite}\end{subarray}}\frac{\Lambda(n)}{n}\ll\frac{1}{\sqrt{N_{0}}}\ll\frac{1}{(1.26)^{k}}.$

Finally, since $|\frac{d}{du}j_{k}(\eta\log u)|=|j_{k-1}(\eta\log
u)-j_{k}(\eta\log u)|\frac{\eta}{u}\leq\frac{\eta}{u}$, partial summation
yields

$\eta\Big{|}\sum_{p\in[N_{0},N_{1}]}\frac{(\chi(p)+\chi_{2}(p)p^{\beta_{1}-1})\log
p}{p^{1+\eta+it}}j_{k}(\eta\log p)\Big{|}\\\
\leq\eta^{2}\int_{N_{0}}^{N_{1}}\Big{|}\sum_{N_{0}\leq p\leq
u}\frac{\chi(p)(1+\chi_{1}(p)p^{\beta_{1}-1})\log
p}{p^{1+i\tau}}\Big{|}\frac{du}{u}+O\Big{(}\frac{\eta}{(1.26)^{k}}\sum_{n=1}^{\infty}\frac{\Lambda(n)}{n^{1+B\eta/2}}\Big{)}\\\
=\eta^{2}\int_{N_{0}}^{N_{1}}\Big{|}\sum_{N_{0}\leq p\leq
u}\frac{\chi(p)(1+\chi_{1}(p)p^{\beta_{1}-1})\log
p}{p^{1+i\tau}}\Big{|}\frac{du}{u}+O\Big{(}\frac{1}{(1.26)^{k}}\Big{)}.$

###  5.4. Finishing the proof

The two preceding subsections show that if H$(\tau,\eta)$ is true, then for
some $k\in[M,2M]$, we have

$\frac{1}{(1.250015191)^{M}}\leq\Big{|}\frac{\eta^{k+1}}{k!}F(s_{0})\Big{|}\leq\eta^{2}\int_{N_{0}}^{N_{1}}\Big{|}\sum_{N_{0}\leq
p\leq u}\frac{\chi(p)(1+\chi_{1}(p)p^{\beta_{1}-1})\log
p}{p^{1+i\tau}}\Big{|}\frac{du}{u}+O\Big{(}\frac{1}{(1.26)^{k}}\Big{)}.$

Since $k\geq M$, it follows from (5.10) and (5.11) that if $K$ is sufficiently
large, then

(5.12) $1\leq
2(1.250015191)^{M}\eta^{2}\int_{N_{0}}^{N_{1}}\Big{|}\sum_{N_{0}\leq p\leq
u}\frac{\chi(p)(1+\chi_{1}(p)p^{\beta_{1}-1})\log
p}{p^{1+i\tau}}\Big{|}\frac{du}{u}.$

We square both sides and apply the Cauchy–Schwarz inequality so that
H$(\eta,\tau)$ implies

(5.13) $1\leq
4(1.250015191)^{2M}\eta^{4}\Big{(}\int_{N_{0}}^{N_{1}}\frac{du}{u}\Big{)}\Big{(}\int_{N_{0}}^{N_{1}}\Big{|}\sum_{N_{0}\leq
p\leq u}\frac{\chi(p)(1+\chi_{1}(p)p^{\beta_{1}-1})\log
p}{p^{1+i\tau}}\Big{|}^{2}\frac{du}{u}\Big{)}\\\
\ll(1.250015191)^{2M}M\eta^{3}\int_{N_{0}}^{N_{1}}\Big{|}\sum_{N_{0}\leq p\leq
u}\frac{\chi(p)(1+\chi_{1}(p)p^{\beta_{1}-1})\log
p}{p^{1+i\tau}}\Big{|}^{2}\frac{du}{u}.$

Note that this identity also holds when $\chi$ is replaced by any Dirichlet
character that it induces, provided that the modulus is changed accordingly.
This is because if $\chi\,(\mathrm{mod}{\,q})$ is induced by
$\chi^{\prime}\,(\mathrm{mod}{\,q^{\prime}})$, then $\chi(p)=\chi^{\prime}(p)$
for all $p>q$, a condition that every $p\in[N_{0},N_{1}]$ satisfies once
$q^{\prime}$ is exchanged for $q$.

Let $\xi=1+10^{-7}$, and define $\sigma$ by the equation $\eta=\xi(1-\sigma)$.
There are $\ll\eta\log(qT)\asymp M$ zeros $\rho=\beta+i\gamma\neq\beta_{1}$ of
$L(s,\chi)$ in the disc $|1+i\tau-\rho|\leq\eta$, and each zero that satisfies
$\beta\geq\sigma$ and $|\gamma|\leq T$ is detected by (5.13) over a
$\tau$-interval of length $\gg\eta$ with $|\tau|\leq T$. Since
$L(s,\chi_{0})\neq 0$ when $\mathrm{Re}(s)>0$ and $|\mathrm{Im}(s)|\leq 14$,
we integrate over $\tau\in[-T,T]$ and conclude that $N_{\chi}(\sigma,T)$ and
$N_{\chi_{1}}^{*}(\sigma,T)$ are

$\ll(1.250015191)^{2M}M^{2}\eta^{2}\int_{N_{0}}^{N_{1}}\int_{-T}^{T}\Big{|}\sum_{N_{0}\leq
p\leq u}\frac{\chi(p)(1+\chi_{1}(p)p^{\beta_{1}-1})\log
p}{p^{1+i\tau}}\Big{|}^{2}\frac{d\tau du}{u}.$

We sum over all characters modulo $q$ and apply the mean value theorem for
integrals to the $u$-integral. This process bounds $N_{q}^{*}(\sigma,T)$ by

(5.14)
$\ll(1.250015191)^{2M}M^{3}\eta\sup_{U\in[N_{0},N_{1}]}\sum_{\chi\,(\mathrm{mod}{\,q})}\int_{-T}^{T}\Big{|}\sum_{N_{0}\leq
p\leq U}\frac{\chi(p)(1+\chi_{1}(p)p^{\beta_{1}-1})\log
p}{p^{1+i\tau}}\Big{|}^{2}d\tau.$

We efficiently estimate (5.14) with a pre-sifted mean value estimate for
Dirichlet polynomials.

######  Lemma 5.6.

Let $\varepsilon>0$ and $T\geq 1$. Let $b(p)$ be complex-valued function
supported on the primes, and suppose that $\sum_{p}|b(p)|$ converges. We have
the bound

$\sum_{\chi\,(\mathrm{mod}{\,q})}\int_{-T}^{T}\Big{|}\sum_{p>(qT)^{1+\varepsilon}}b(p)\chi(p)p^{-i\tau}\Big{|}^{2}d\tau\ll_{\varepsilon}\frac{1}{\log(qT)}\sum_{p>(qT)^{1+\varepsilon}}|b(p)|^{2}p.$

###### Proof.

Let $\varepsilon>0$, and choose $b(p)$ to be supported on primes
$p\geq(qT)^{1+\varepsilon}$. Let $z\geq\frac{3}{2}$. It follows from
Plancherel’s theorem that

(5.15)
$\sum_{\chi\,(\mathrm{mod}{\,q})}\int_{-T}^{T}\Big{|}\sum_{p>z}b(p)\chi(p)p^{-i\tau}\Big{|}^{2}d\tau\ll
T^{2}\int_{0}^{\infty}\sum_{\chi\,(\mathrm{mod}{\,q})}\Big{|}\sum_{\begin{subarray}{c}y<p\leq
ye^{1/T}\\\ p>z\end{subarray}}b(p)\chi(p)\Big{|}^{2}\frac{dy}{y}.$

(See [3, Theorem 1].) By the orthogonality of Dirichlet characters and the
Cauchy–Schwarz inequality, the right hand side of (5.15) equals

$T^{2}\int_{0}^{\infty}\varphi(q)\sum_{\begin{subarray}{c}1\leq a\leq q\\\
\gcd(a,q)=1\end{subarray}}\Big{|}\sum_{\begin{subarray}{c}p\equiv
a\,(\mathrm{mod}{\,q})\\\ y<p\leq ye^{1/T}\\\
p>z\end{subarray}}b(p)\Big{|}^{2}\frac{dy}{y}\\\ \ll
T^{2}\int_{0}^{\infty}\varphi(q)\sum_{\begin{subarray}{c}1\leq a\leq q\\\
\gcd(a,q)=1\end{subarray}}\Big{(}\sum_{\begin{subarray}{c}p\equiv
a\,(\mathrm{mod}{\,q})\\\ y<p\leq ye^{1/T}\\\
p>z\end{subarray}}|b(p)|^{2}\Big{)}\Big{(}\sum_{\begin{subarray}{c}p\equiv
a\,(\mathrm{mod}{\,q})\\\ y<p\leq ye^{1/T}\\\
p>z\end{subarray}}1\Big{)}\frac{dy}{y}.$

We apply the Selberg sieve [11, Proof of Thm 21.4] and bound (5.15) by

$\ll T^{2}\int_{0}^{\infty}\varphi(q)\sum_{\begin{subarray}{c}1\leq a\leq q\\\
\gcd(a,q)=1\end{subarray}}\Big{(}\sum_{\begin{subarray}{c}p\equiv
a\,(\mathrm{mod}{\,q})\\\ y<p\leq ye^{1/T}\\\
p>z\end{subarray}}|b(p)|^{2}\Big{)}\Big{(}\frac{y(e^{1/T}-1)}{\varphi(q)\log
z}+z^{2}\Big{)}\frac{dy}{y}\\\ \ll\int_{0}^{\infty}\Big{(}\frac{T}{\log
z}+\frac{\varphi(q)T^{2}z^{2}}{y}\Big{)}\Big{(}\sum_{\begin{subarray}{c}y<p\leq
ye^{1/T}\\\ p>z\end{subarray}}|b(p)|^{2}\Big{)}dy=\frac{1}{\log
z}\sum_{p>z}|b(p)|^{2}p\Big{(}1+\frac{\varphi(q)Tz^{2}\log z}{p}\Big{)}.$

We choose $z=(qT)^{\varepsilon/3}$ so that $\varphi(q)Tz^{2}\log
z\ll_{\varepsilon}(qT)^{1+\varepsilon}$. The desired result follows. ∎

######  Corollary 5.7.

Let $T\geq 1$. Recall the definition of $\nu(u)$ from Theorem 2.1. If
$\phi\geq\frac{1}{73.4}$ in Lemma 5.5 and $U\in[N_{0},N_{1}]$, then

$\sum_{\chi\,(\mathrm{mod}{\,q})}\int_{-T}^{T}\Big{|}\sum_{N_{0}\leq p\leq
U}\frac{\chi(p)\log
p}{p^{1+i\tau}}(1+\chi_{1}(p)p^{\beta_{1}-1})\Big{|}^{2}d\tau\ll\nu(qT)\log
qT\ll\nu(qT)\frac{M}{\eta}.$

###### Proof.

If $\phi\geq\frac{1}{73.4}$ in Lemma 5.5, then $N_{0}\geq(qT)^{1.00004}$. Let
$U\in[N_{0},N_{1}]$, and choose $b(p)=\frac{\log
p}{p}(1+\chi_{1}(p)p^{\beta_{1}-1})$ when $p\in[N_{0},U]$ and $b(p)=0$
otherwise. The left hand side of Lemma 5.6 is then

$\ll\frac{1}{\log(qT)}\sum_{N_{0}\leq p\leq U}\frac{(\log
p)^{2}}{p}(1+\chi_{1}(p)p^{\beta_{1}-1})^{2}\ll\sum_{N_{0}\leq p\leq
N_{1}}\frac{\log p}{p}(1+\chi_{1}(p)p^{\beta_{1}-1})^{2}.$

By arguments essentially identical to those in [1, Lemme C], this is bounded
as desired. ∎

In light of (5.1), (5.2), and (5.11), it follows from (5.14) and Corollary 5.7
that

(5.16)
$N_{q}^{*}(\sigma,T)\ll\nu(qT)(1.250015191)^{2M}M^{4}\ll\nu(qT)(qT)^{112.20562143\max\\{\frac{1}{73.4},\phi\\}\eta}.$

Since $\eta=\xi(1-\sigma)$, the constraints in (5.2) imply that

(5.17)
$N_{q}^{*}(\sigma,T)\ll\nu(qT)(qT)^{112.20563265056215\max\\{\frac{1}{73.4},\phi\\}(1-\sigma)},\quad
1-\frac{1}{2A\xi}\leq\sigma<1-\frac{1}{\xi\log(qT)}.$

If $\sigma\geq 1-\frac{1}{\xi\log(qT)}$ or if $T\leq 100$, then the bound on
$N_{q}^{*}(\sigma,T)$ holds trivially. With our choices for $A$ and $\xi$, we
conclude that

(5.18)
$N_{q}^{*}(\sigma,T)\ll\nu(qT)(qT)^{112.20563265056215\max\\{\frac{1}{73.4},\phi\\}(1-\sigma)},\qquad
3/5\leq\sigma\leq 1.$

It remains to choose $\phi=\frac{1}{6}+10^{-7}$ in (5.8), which is permissible
by recent work of Petrow and Young [17]. With this choice, (5.18) holds
trivially for $1/2\leq\sigma\leq 3/5$ because of the trivial bound
$N_{q}^{*}(\sigma,T)\ll\varphi(q)T\log(qT)$. This completes the proof of
Theorem 2.1.

######  Remark 5.8.

The exponent $112.20563265056215\max\\{\frac{1}{73.4},\phi\\}$ in (5.18) is
nearly the limit of this approach in view of Remark 5.3 and the footnote on
page 2 above. It is classical that the generalized Lindelöf hypothesis (that
(5.8) holds for any fixed $\phi>0$) implies the density hypothesis: For all
$\varepsilon>0$ and all $1/2\leq\sigma\leq 1$, we have
$N_{q}(\sigma,T)\ll_{\varepsilon}(qT)^{2(1-\sigma)+\varepsilon}$. If (5.8)
holds for any fixed $0<\phi\leq\frac{1}{57}$, then in the range
$3/5\leq\sigma\leq 1$, our above bound for $N_{q}^{*}(\sigma,T)$ surpasses the
density hypothesis while incorporating zero repulsion.

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