Weighted Prime Number Theorem on Arithmetic Progressions with Refinements

1. Introduction

We define the weighted counting function of primes in arithmetic progressions as

$${\pi }_w(x;q,a)=\sum _{\genfrac{}{}{0.0pt}{}{p<x:\mathrm{prime}}{p\equiv a(modq)}}{p}^w(w\ge 0)$$

for $q\in \mathbb{Z}$. This is a generalization of the classical counting function $\pi (x;q,a)={\pi }_0(x;q,a)$. In 1837 Dirichlet proved that

$$\pi (x;q,a)\sim \pi (x;q,b)(x\to \infty )$$

for any a, b which are coprime with q, where $f\left(x\right)\sim g\left(x\right)$$(x\to \infty )$ means that $f\left(x\right)/g\left(x\right)\to 1$ as $x\to \infty$.

On the other hand, in 1853 Chebyshev found the phenomenon that there tend to be more primes p satisfying $p\equiv 3(mod4)$ than those with $p\equiv 1(mod4)$. In fact, the inequality

$$\pi (x;4,3)\ge \pi (x;4,1)$$

(1.1)

holds for any x less than 26861, which is the first prime number violating the inequality (1.1). However, the both sides draw equal at the next prime 26863, and $\pi (x;4,3)$ gets ahead again until 616841. It is computed that more than 97% of $x<{10}^{11}$ satisfy the inequality (1.1). This phenomenon is called Chebyshev’s bias, and is one of major unsolved mysteries in number theory.

Later, Littlewood [1] proved that the difference $\pi (x;4,3)-\pi (x;4,1)$ changes its sign infinitely many times. In 2023, Aoki and Koyama [2] suggested that Littlewood’s phenomenon no longer holds if we put the weight $p^{-1/2}$. They actually obtained the asymptotic under the assumption of the Deep Riemann Hypothesis that

$${\pi }_{\frac{1}{2}}(x;4,3)-{\pi }_{\frac{1}{2}}(x;4,1)\sim \frac{1}{2}loglogx(x\to \infty ).$$

(1.2)

The Deep Riemann Hypothesis (DRH) is a conjecture asserting the convergence of the Euler product of the corresponding L-function on the critical line. More generally they defined Chebyshev bias towards $p\equiv b(modq)$ (or against $p\equiv a(modq)$) as the asymptotic

$${\pi }_{\frac{1}{2}}(x;q,b)-{\pi }_{\frac{1}{2}}(x;q,a)\sim Cloglogx(x\to \infty )$$

with a constant $C>0$ depending on q. Under DRH, it is also proved for any $w>1/2$ that

$${\pi }_w(x;q,b)-{\pi }_w(x;q,q)=O\left(1\right)(x\to \infty ).$$

(1.3)

This was one of the reasons why Aoki-Koyama chose $w=1/2$ for formulating the bias. But the behavior of the left hand side for $0<w<1/2$ was unknown.

In this paper, we will solve this problem by generalizing Littlewood’s theorem to $0\le w<1/2$. In what follows we use the standard notation

$$Li\left(x\right)={\int }_2^x{\displaystyle \frac{dt}{logt}}.$$

and $\phi \left(q\right)=\#{(\mathbb{Z}/q\mathbb{Z})}^{\times }$. Our first main theorem is as follows.

Theorem 1.1

(Weighted prime number theorem). Let $a\in \mathbb{Z}$ be coprime with $q\in \mathbb{Z}$. For $0\le w<1$, it holds that

$${\pi }_w(x;q,a)\sim {\displaystyle \frac{1}{\phi \left(q\right)}}Li\left(x^{1-w}\right)(x\to \infty ).$$

We write $f\left(x\right)=\Omega \left(g\right(x\left)\right)$$(x\to \infty )$ if the following holds: There exists a constant $C>0$ such that for any $x_0>0$, there exist $x_1,x_2>x_0$ such that $f\left(x_1\right)>Cg\left(x_1\right)$ and $f\left(x_2\right)<-Cg\left(x_2\right)$.

It is known by Stark ([3] Theorem 2) that for any coprime $a,b$$(a\ne b)$ with q

$$\pi (x;q,a)-\pi (x;q,b)=\Omega \left({\displaystyle \frac{x^{\frac{1}{2}}}{logx}}\right)(x\to \infty )$$

(1.4)

under the assumption that the Dirichlet L-function $L(s,\chi )$ has no zeros in the interval $0<s<1$ for any Dirichlet character $\chi$ modulo q. Actually Stark obtained the estimate (1.4) for more general cases $\phi \left(q\right)\pi (x;q,a)-\phi \left(q^{\prime }\right)\pi (x;q^{\prime },b)$, but for our purpose the case $q=q^{\prime }$ is sufficient.

Our second main theorem is a generalization of it.

Theorem 1.2.

Suppose that $L(s,\chi )$ has no zeros in the interval $0<s<1$ for any Dirichlet character $\chi$ modulo q. For $0\le w<1/2$ and any coprime $a,b$$(a\ne b)$ with q, it holds that

$${\pi }_w(x;q,a)-{\pi }_w(x;q,b)=\Omega \left({\displaystyle \frac{x^{\frac{1}{2}-w}}{logx}}\right)(x\to \infty ).$$

In particular, ${\pi }_w(x;q,a)-{\pi }_w(x;q,b)$$(0\le w<1/2)$ changes its sign infinitely many times.

Theorem 1.2 together with the estimate (1.3) justifies the choice of $w=1/2$ for the formulation of Chebyshev’s bias.

2. Weighted Prime Number Theorem

The following lemma was proved in the previous paper [4].

Lemma 2.1.

Assume $f\left(x\right)$ and $g\left(x\right)$ are positive valued function on $x\ge 0$. If ${\displaystyle \underset{x\to \infty }{lim}{\int }_0^{x}f\left(t\right)dt=\infty }$ and $g\left(t\right)=o\left(1\right)$, it holds that

$${\int }_0^{x}f\left(t\right)g\left(t\right)dt=o\left({\int }_0^{x}f\left(t\right)dt\right)(x\to \infty ).$$

For q, $a\in \mathbb{Z}$ with $(a,q)=1$ and $0\le w<1$, we denote

$${\psi }_w(x;q,a)=\sum _{\begin{array}{c}p\equiv a(modq)\\ p^m\le x\end{array}}{\displaystyle \frac{logp}{p^{mw}}},$$

where the sum is taken over pairs $(p,m)$ with m a positive integer and p a prime satisfying $p\equiv a(modq)$ and $p^m\le x$.

Lemma 2.2.

$${\psi }_w(x;q,a)\sim {\displaystyle \frac{x^{1-w}}{\phi \left(q\right)(1-w)}}(x\to \infty )$$

Proof. 

Putting $\psi (x;q,a)={\psi }_0(x;q,a)$, we have by partial summation that

$${\psi }_w(x;q,a)={\displaystyle \frac{\psi (x;q,a)}{x^w}}+w{\int }_2^x{\displaystyle \frac{\psi (t;q,a)}{t^{w+1}}}dt.$$

From the classical prime number theorem $\psi (x;q,a)=x/\phi \left(q\right)+o\left(x\right)$$(x\to \infty )$, we compute

$${\psi }_w(x;q,a)={\displaystyle \frac{1}{\phi \left(q\right)}}\left(x^{1-w}+o\left(x^{1-w}\right)+w{\int }_2^x{\displaystyle \frac{1}{t^w}}dt+{\int }_2^{x}o\left({\displaystyle \frac{1}{t^w}}\right)dt\right).$$

Here the last integral is estimated by Lemma 2.1 as

$${\int }_2^{x}o\left({\displaystyle \frac{1}{t^w}}\right)dt=o\left({\int }_2^x{\displaystyle \frac{1}{t^w}}dt\right).$$

Thus we have

$$\begin{array}{cc}\hfill {\psi }_w(x;q,a)& ={\displaystyle \frac{1}{\phi \left(q\right)}}\left(x^{1-w}+o\left(x^{1-w}\right)+w{\int }_2^x{\displaystyle \frac{1}{t^w}}dt+o\left({\int }_2^x{\displaystyle \frac{1}{t^w}}dt\right)\right)\hfill \\ \hfill & ={\displaystyle \frac{1}{\phi \left(q\right)}}{\displaystyle \frac{x^{1-w}}{1-w}}+o\left(x^{1-w}\right).\hfill \end{array}$$

□

For $0\le w<1$ we put

$${\theta }_w(x;q,a)=\sum _{\begin{array}{c}p\equiv a(modq)\\ p\le x\end{array}}{\displaystyle \frac{logp}{p^w}},$$

where the sum is taken over primes p satisfying $p\equiv a(modq)$ and $p\le x$.

Lemma 2.3.

It holds that

$${\psi }_w(x;q,a)-{\displaystyle \frac{\sqrt{x}logx}{(1-2w)x^w}}\le {\theta }_w(x;q,a)\le {\psi }_w(x;q,a).$$

Proof. 

From the definitions the right iequality is immediate. The left one is deduced as follows.

$$\begin{array}{cc}\hfill {\psi }_w(x;q,a)-{\theta }_w(x;q,a)& \le logx\sum _{\begin{array}{c}p\equiv a(modq)\\ p\le \sqrt{x}\end{array}}{\displaystyle \frac{1}{p^{2w}}}\le logx\sum _{2\le n\le \sqrt{x}}{\displaystyle \frac{1}{n^{2w}}}\hfill \\ \hfill & \le logx{\int }_1^{\sqrt{x}}{\displaystyle \frac{dt}{t^{2w}}}\le {\displaystyle \frac{x^{\frac{1}{2}-w}logx}{1-2w}}.\hfill \end{array}$$

□

Proof of Theorem 1.1. 

From Lemmas 2.2 and 2.3 we have

$$\begin{array}{c}\hfill {\theta }_w(x;q,a)\sim {\psi }_w(x;q,a)\sim {\displaystyle \frac{x^{1-w}}{\phi \left(q\right)(1-w)}}(x\to \infty ).\end{array}$$

(2.1)

Now we will establish the relation between ${\theta }_w(x;q,a)$ and ${\pi }_w(x;q,a)$. Clearly it holds that

$$\begin{array}{cc}\hfill {\theta }_w(x;q,a)& \le \sum _{\begin{array}{c}p\equiv a(modq)\\ p\le \sqrt{x}\end{array}}{\displaystyle \frac{logx}{p^w}}=(logx){\pi }_w(x;q,a).\hfill \end{array}$$

Hence

$${\displaystyle \frac{{\theta }_w(x;q,a)}{logx}}\le {\pi }_w(x;q,a).$$

On the other hand, for $0\le \epsilon <1$ we have $1<x^{1-\epsilon }\le x$ and

$$\begin{array}{cc}\hfill {\theta }_w(x;q,a)\ge \sum _{\begin{array}{c}p\equiv a(modq)\\ x^{1-\epsilon }\le p\le x\end{array}}{\displaystyle \frac{logp}{p^w}}& \ge \sum _{\begin{array}{c}p\equiv a(modq)\\ x^{1-\epsilon }\le p\le x\end{array}}{\displaystyle \frac{log{x}^{1-\epsilon }}{p^w}}\hfill \\ \hfill & =log{x}^{1-\epsilon }\sum _{\begin{array}{c}p\equiv a(modq)\\ x^{1-\epsilon }\le p\le x\end{array}}{\displaystyle \frac{1}{p^w}}\hfill \\ \hfill & =(log{x}^{1-\epsilon })({\pi }_w(x;q,a)-{\pi }_w(x^{1-\epsilon };q,a)).\hfill \end{array}$$

Therefore

$$\begin{array}{c}\hfill {\pi }_w(x;q,a)\le {\displaystyle \frac{1}{1-\epsilon }}·{\displaystyle \frac{{\theta }_w(x;q,a)}{logx}}+{\pi }_w(x^{1-\epsilon };q,a).\end{array}$$

(2.2)

Since the last term is estimated as

$$\begin{array}{cc}\hfill {\pi }_w(x^{1-\epsilon };q,a)=\sum _{\begin{array}{c}p\equiv a(modq)\\ p\le x^{1-\epsilon }\end{array}}{\displaystyle \frac{1}{p^w}}& <\sum _{n\le x^{1-\epsilon }}{\displaystyle \frac{1}{n^w}}=O\left({\left(x^{1-w}\right)}^{1-\epsilon }\right),\hfill \end{array}$$

We conclude that

$$\begin{array}{c}\hfill {\displaystyle \frac{{\theta }_w(x;q,a)}{logx}}\le {\pi }_w(x;q,a)\le {\displaystyle \frac{1}{1-\epsilon }}·{\displaystyle \frac{{\theta }_w(x;q,a)}{logx}}+O\left({\left(x^{1-\epsilon }\right)}^{1-w}\right).\end{array}$$

(2.3)

Now combining with (2.1) we reach

$${\pi }_w(x;q,a)\sim {\displaystyle \frac{{\theta }_w(x;q,a)}{logx}}\sim {\displaystyle \frac{x^{1-w}}{\phi \left(q\right)(1-w)logx}}\sim {\displaystyle \frac{\mathrm{Li}\left(x^{1-w}\right)}{\phi \left(q\right)}}.$$

□

3. Refinements

In this section we prove Threom 1.2. The following lemma plays the role for reducing the genral weight w to the case of $w=0$.

Lemma 3.1.

$${\int }_0^{\infty }{\pi }_w(e^u;q,a)e^{-us}du={\displaystyle \frac{s+w}{s}}{\int }_0^{\infty }\pi (e^u;q,a)e^{-u(s+w)}du.$$

Proof. 

Denote by $p_n$ the n-th smallest element in the set of primes p such that $p\equiv a(modq)$. Putting $x=e^u$, we compute

$$\begin{array}{cc}\hfill {\int }_0^{\infty }{\pi }_w(e^u;q,a)e^{-us}du& ={\int }_1^{\infty }{\pi }_w(x;q,a)x^{-s-1}dx\hfill \\ \hfill & =\sum _{n=1}^{\infty }{\int }_{p_n}^{p_{n+1}}\sum _{\begin{array}{c}p\equiv a(modq)\\ p\le p_n\end{array}}{p}^{-w}{x}^{-s-1}dx\hfill \\ \hfill & =\sum _{n=1}^{\infty }\sum _{\begin{array}{c}p\equiv a(modq)\\ p\le p_n\end{array}}{p}^{-w}{\displaystyle \frac{p_n^{-s}-p_{n+1}^{-s}}{s}}\hfill \\ \hfill & ={\displaystyle \frac{1}{s}}\left(\sum _{n=1}^{\infty }\sum _{\begin{array}{c}p\equiv a(modq)\\ p\le p_n\end{array}}{p}^{-w}{p}_n^{-s}-\sum _{n=2}^{\infty }\sum _{\begin{array}{c}p\equiv a(modq)\\ p\le p_{n-1}\end{array}}{p}^{-w}{p}_n^{-s}\right)\hfill \\ \hfill & ={\displaystyle \frac{1}{s}}\left(p_1^{-s-w}+\sum _{n=2}^{\infty }{p}_n^{-s-w}\right)\hfill \\ \hfill & ={\displaystyle \frac{1}{s}}\sum _{n=1}^{\infty }{p}_n^{-s-w}.\hfill \end{array}$$

Hence for $w=0$ we have

$${\int }_0^{\infty }\pi (e^u;q,a)e^{-us}du={\displaystyle \frac{1}{s}}\sum _{n=1}^{\infty }{p}_n^{-s}.$$

By substituting $s+w$ in the variable s, we obtain

$$\begin{array}{cc}\hfill {\int }_0^{\infty }\pi (e^u;q,a)e^{-u(s+w)}du& ={\displaystyle \frac{1}{s+w}}\sum _{n=1}^{\infty }{p}_n^{-s-w}\hfill \\ \hfill & ={\displaystyle \frac{s}{s+w}}{\int }_0^{\infty }{\pi }_w(e^u;q,a)e^{-us}du.\hfill \end{array}$$

□

We will apply the theory of Tauberian theorems, which was developed by Ingham in the following theorem.

Theorem 3.2. ([5] Theorem 1) Let

$$F\left(s\right)={\int }_0^{\infty }A\left(u\right)e^{-su}du,$$

where $A\left(u\right)$ is real valued and absolutely integrable on every interval $0\le u\le U$, and the integral is absolutely convergent in some half plane $\sigma >{\sigma }_1\ge 0$.

Let $A^{*}\left(u\right)$ be a real trigonometrical polynomial

$$A^{*}\left(u\right)=\sum _{n=-N}^N{\alpha }_n{e}^{i{\gamma }_{n}u}({\gamma }_n\in \mathbb{R},{\gamma }_n={\gamma }_{-n},{\alpha }_n=\overline{{\alpha }_{-n}}),$$

and let

$$F^{*}\left(s\right)={\int }_0^{\infty }{A}^{*}\left(u\right)e^{-su}du=\sum _{n=-N}^N{\displaystyle \frac{{\alpha }_n}{s-i{\gamma }_n}}(\sigma >0).$$

Suppose that $F\left(s\right)-F^{*}\left(s\right)$ is regular in the region $\sigma \ge 0$, $-T\le t\le T$ for some $T>0$.

Then, it holds that

$$\underset{u\to \infty }{lim\; inf}A\left(u\right)\le \underset{u\to \infty }{lim}{A}_T^{*}\left(u\right)\le \underset{u\to \infty }{lim\; sup}A\left(u\right),$$

where

$$\begin{array}{cc}\hfill A_T^{*}\left(u\right)& =\sum _{|{\gamma }_n|\le T}\left(1-{\displaystyle \frac{|{\gamma }_n|}{T}}\right){\alpha }_n{e}^{i{\gamma }_{n}u}\hfill \\ \hfill & ={\alpha }_0+2\mathrm{Re}\left(\sum _{0<{\gamma }_n<T}\left(1-{\displaystyle \frac{{\gamma }_n}{T}}\right){\alpha }_n{e}^{i{\gamma }_{n}u}\right).\hfill \end{array}$$

This theorem was reformulated by Stark for the purpose of refining the estimate of $\pi (x;q,a)-\pi (x;q,b)$ as follows:

Theorem 3.3. ([3] p.314) Put $s=\sigma +it\in \mathbb{C}$. Let

$${\displaystyle \frac{F\left(s\right)}{s}}={\int }_0^{\infty }A\left(u\right)e^{-su}du,$$

where $A\left(u\right)$ is real valued and absolutely integrable on every interval $0\le u\le U$, and the integral is absolutely convergent for $\sigma >1$. For any real sequence ${\gamma }_n$$(n\in \mathbb{Z})$ with ${\gamma }_{-n}={\gamma }_n$ and coefficients ${\alpha }_n\in \mathbb{C}$ with ${\alpha }_{-n}=\overline{{\alpha }_n}$, set

$$F_1\left(s\right)=\sum _{n=-N}^N{\alpha }_{n}log\left(s-{\displaystyle \frac{1}{2}}-i{\gamma }_n\right).$$

Suppose that for some $T>0$, $F^{\prime }\left(s\right)-F_1^{\prime }\left(s\right)$ is continuous in the region $\sigma \ge 1/2$, $-T\le t\le T$ and analytic in the interior of this region. Then for any $u_0$,

$$\underset{u\to \infty }{lim\; sup}{\displaystyle \frac{uA\left(u\right)}{e^{u/2}}}\ge -\sum _{|{\gamma }_n|\le T}{\displaystyle \frac{{\alpha }_n}{\frac{1}{2}+i{\gamma }_n}}\left(1-{\displaystyle \frac{|{\gamma }_n|}{T}}\right)e^{i{\gamma }_n{u}_0}.$$

Theorem 3.3 follows from Theorem 3.2 applied to the derivative with respect to s of

$$\begin{array}{cc}\hfill {\displaystyle \frac{F\left(s\right)}{s}}& ={\int }_1^{\infty }\left(A\left(u\right)+{\displaystyle \frac{1}{u}}\sum _{n=-N}^N{\displaystyle \frac{{\alpha }_n}{\frac{1}{2}+i{\gamma }_n}}{e}^{({\displaystyle \frac{1}{2}}+i{\gamma }_n)u}\right)e^{-su}du.\hfill \\ \hfill & =\sum _{n=-N}^N{\displaystyle \frac{{\alpha }_n}{\frac{1}{2}+i{\gamma }_n}}log\left(s-{\displaystyle \frac{1}{2}}-i{\gamma }_n\right)+\left(\mathrm{an}\mathrm{entire}\mathrm{function}\mathrm{in}s\right)\hfill \end{array}$$

(3.1)

The estimate (1.4) is an immediate consequence of Theorem 3.3 and the fact that the function

$${\int }_0^{\infty }\left(\pi (e^u;q,a)-\pi (e^u;q,b)\right)e^{-us}du$$

is analytic in $\mathrm{Re}\left(s\right)>1/2$ ([3] Lemmas 1 and 2).

Proof of Theorem 1.2. 

From Lemma 3.1 and the above discussion, the function

$${\int }_0^{\infty }\left({\pi }_w(e^u;q,a)-{\pi }_w(e^u;q,b)\right)e^{-us}du$$

is analytic in $\mathrm{Re}\left(s\right)>{\displaystyle \frac{1}{2}}-w$. For $0<w<{\displaystyle \frac{1}{2}}$, we replace (3.1) with the following function:

$$\begin{array}{cc}\hfill {\displaystyle \frac{F_w\left(s\right)}{s+w}}& =\sum _{n=-N}^N{\displaystyle \frac{{\alpha }_n}{\frac{1}{2}+i{\gamma }_n}}log\left(s+w-{\displaystyle \frac{1}{2}}-i{\gamma }_n\right)+\left(\mathrm{an}\mathrm{entire}\mathrm{function}\mathrm{in}s\right)\hfill \\ \hfill & ={\int }_1^{\infty }\left(A_w\left(u\right)+{\displaystyle \frac{1}{u}}\sum _{n=-N}^N{\displaystyle \frac{{\alpha }_n}{\frac{1}{2}+i{\gamma }_n}}{e}^{({\displaystyle \frac{1}{2}}-w+i{\gamma }_n)u}\right)e^{-su}du,\hfill \end{array}$$

where $A_w\left(u\right)={\pi }_w(e^u;q,a)-{\pi }_w(e^u;q,b)$. Then for any $u_0$,

$$\underset{u\to \infty }{lim\; sup}{\displaystyle \frac{u{A}_w\left(u\right)}{e^{(\frac{1}{2}-w)u}}}\ge -\sum _{|{\gamma }_n|\le T}{\displaystyle \frac{{\alpha }_n}{\frac{1}{2}+i{\gamma }_n}}\left(1-{\displaystyle \frac{|{\gamma }_n|}{T}}\right)e^{i{\gamma }_n{u}_0}.$$

Consequently it holds that

$$A_w\left(u\right)=\Omega \left({\displaystyle \frac{e^{(\frac{1}{2}-w)u}}{u}}\right),$$

which is the desired result. □