A Conditional Explicit Result for the Prime Number Theorem in Short Intervals

TL;DR

This paper provides an explicit bound for the prime number theorem in short intervals assuming the Riemann hypothesis, advancing understanding of prime distribution in small ranges.

Contribution

It introduces a new explicit bound for primes in short intervals under the Riemann hypothesis, improving previous estimates.

Findings

Explicit bound for primes in short intervals

Assumption of Riemann hypothesis used

Enhanced understanding of prime distribution

Abstract

We furnish an explicit bound for the prime number theorem in short intervals on the assumption of the Riemann hypothesis.

Equations

$$\Lambda(n)=\left\{\begin{array}[]{ll}\log p&:\hskip 7.22743ptn=p^{m},\text{ $p$ is prime, $m\in\mathbb{N}$}\\ 0&:\hskip 7.22743pt\text{otherwise,}\end{array}\right.$$

$$\psi (x+h)-\psi (h)\sim h$$

$$|\psi(x+h)-\psi(x)-h|<\frac{1}{\pi}\sqrt{x}\log x\log\bigg{(}\frac{h}{\sqrt{x}\log x}\bigg{)}+2\sqrt{x}\log x.$$

$$|\psi (x+h)-\psi (x)-h|\ll \sqrt{x}\log x\log \left(f(x)\right)=o(h).$$

$$|\psi (x)-x|<\frac{1}{8\pi }\sqrt{x}{\log }^{2}x.$$

$$|\psi (x+h)-\psi (x)-h|<\frac{1}{4\pi }\sqrt{x+h}{\log }^2(x+h).$$

$$\psi (x)=x-\sum _{\rho }\frac{x^{\rho }}{\rho }-\log 2\pi -\frac{1}{2}\log (1-x^{-2}),$$

$${\psi }_1(x)=\sum _{n\le x}(x-n)\Lambda (n)={\int }_2^x\psi (t)dt$$

$${\psi }_1(x)=\frac{x^2}{2}-\sum _{\rho }\frac{x^{\rho +1}}{\rho (\rho +1)}-x\log (2\pi )+ϵ(x)$$

$$1.545<ϵ(x)<\mathrm{2.069.}$$

$$ϵ(x)$$

$$<2\log 2\pi -2+2^{\frac{3}{2}}(\gamma +2-\log 4\pi )+\log \frac{3\sqrt{3}}{4}<2.069$$

$$ϵ(x)>2\log 2\pi -2-2^{\frac{3}{2}}(\gamma +2-\log 4\pi )>\mathrm{1.545.}$$

$$w(n)=\left\{\begin{array}[]{ll}(n-x+\Delta)/\Delta&:\hskip 7.22743ptx-\Delta\leq n\leq x\\ 1&:\hskip 7.22743ptx\leq n\leq x+h\\ (x+h+\Delta-n)/\Delta&:\hskip 7.22743ptx+h\leq n\leq x+h+\Delta\\ 0&:\hskip 7.22743pt\text{otherwise.}\end{array}\right.$$

$$\sum _n\Lambda (n)w(n)$$

$$\sum _n\Lambda (n)w(n)=h+\Delta -\frac{1}{\Delta }\sum _{\rho }S(\rho )+ϵ(\Delta )$$

$$S(\rho )=\frac{(x+h+\Delta {)}^{\rho +1}-(x+h{)}^{\rho +1}-x^{\rho +1}+(x-\Delta {)}^{\rho +1}}{\rho (\rho +1)}$$

$$|ϵ(\Delta )|<\frac{21}{20\Delta }.$$

$$\sum_{\rho}S(\rho)=\bigg{(}\sum_{|\gamma|\leq\alpha x/h}+\sum_{\alpha x/h<|\gamma|<\beta x/\Delta}+\sum_{|\gamma|\geq\beta x/\Delta}\bigg{)}S(\rho)$$

$$\left|\sum _{|\gamma |\ge \beta x/\Delta }S(\rho )\right|<\frac{4\Delta (x+h+\Delta {)}^{3/2}}{\pi \beta x}\log (\beta x/\Delta )$$

$$|S(\rho )|\le \frac{4(x+h+\Delta {)}^{3/2}}{{\gamma }^2}.$$

$$\sum _{\gamma \ge T}\frac{1}{{\gamma }^2}<\frac{1}{2\pi }\frac{\log T}{T}.$$

$$P(T):=\frac{T}{2\pi }\log \frac{T}{2\pi }-\frac{T}{2\pi }+\frac{7}{8}$$

$$|Q(T)|\le R(T)=a_1\log T+a_2\log \log T+a_3$$

$$\left|\sum _{|\gamma |\le \alpha x/h}S(\rho )\right|<\frac{\alpha x(h+\Delta )\Delta }{\pi h\sqrt{x-\Delta }}\log (\alpha x/h).$$

$$S(\rho )={\int }_{x+h}^{x+h+\Delta }{\int }_{u-h-\Delta }^u{t}^{\rho -1}dtdu,$$

$$|S(\rho )|<\frac{(h+\Delta )\Delta }{\sqrt{x-\Delta }}.$$

$$N(T)<\frac{T\log T}{2\pi },$$

$$\sum _{T_1<\gamma <T_2}\frac{1}{\gamma }$$

$$|E(T)|\le \frac{2A_1\log T+2A_0+A_1+A_2}{T^2}$$

$$\left|\sum_{\alpha x/h<|\gamma|<\beta x/\Delta}S(\rho)\,\right|<\Delta(x+h+\Delta)^{1/2}\bigg{(}\frac{1}{\pi}\log\bigg{(}\frac{\beta h}{\alpha\Delta}\bigg{)}\log\bigg{(}\frac{\alpha\beta x^{2}}{4\pi^{2}h\Delta}\bigg{)}+5.4\bigg{)}.$$

Full text

1 Introduction

The von Mangoldt function is defined as

[TABLE]

and we will consider the sum $\psi(x)=\sum_{n\leq x}\Lambda(n)$. The prime number theorem (PNT) is the statement $\psi(x)\sim x$ as $x\rightarrow\infty$. For the PNT in short intervals, it is known that

[TABLE]

provided that $h$ grows suitably with respect to $x$. Heath-Brown [9] has shown that one can take $h=x^{\frac{7}{12}-\epsilon}$ provided that $\epsilon\rightarrow 0$ as $x\rightarrow\infty$. Assuming the Riemann hypothesis (RH), Selberg [14] showed that (1) is true for any $h=h(x)$ such that $h/(x^{1/2}\log x)\rightarrow\infty$ as $x\rightarrow\infty$. On the other hand, Maier [11] has shown that the statement is false for $h=(\log x)^{\lambda}$ for any $\lambda>1$.

In this paper we prove the following explicit version of Selberg’s result.

Theorem 1**.**

Assuming RH, for any $h$ satisfying $\sqrt{x}\log x\leq h\leq x^{\frac{3}{4}}$ and all $x\geq e^{10}$ we have

[TABLE]

Selberg’s result follows from Theorem 1 for any $h=f(x)\sqrt{x}\log x$ with unbounded $f(x)=o(x)$, in that we would have

[TABLE]

For $h=c\sqrt{x}\log x$, Theorem 1 implies Cramér’s [6] result on primes in the interval $(x,x+h)$ for all sufficiently large $x$ and $c$. In an earlier paper [7], the author showed that $c=1+\epsilon$ is suitable for any $\epsilon>0$ and for all sufficiently large $x$. Carneiro, Milinovich and Soundararajan [4] have since shown that we can take $c=22/55$ for all $x\geq 4$. The same methods used in [7] are applied to reach Theorem 1. As such, it could be possible to sharpen Theorem 1 using the techniques in [4].

The closest result to Theorem 1 is the following from Schoenfeld [13].

Theorem 2**.**

Assuming RH, for $x\geq 73.2$ we have

[TABLE]

Schoenfeld’s result confirms Selberg’s theorem for the slightly stronger condition of $h/(\sqrt{x}\log^{2}x)\rightarrow\infty$. One also has from the above

[TABLE]

When $x$ is sufficiently large, Theorem 1 improves the leading constant in this bound for any choice of $h\leq x^{0.735}$.

2 Proof of Theorem 1

2.1 A smooth explicit formula

The Riemann–von Mangoldt explicit formula relates $\psi(x)$ to the zeros of the Riemann zeta-function $\zeta(s)$ (e.g. see Ingham [10]). Tor all non-integer $x>0$,

[TABLE]

where the sum is over all non-trivial zeroes $\rho=\beta+i\gamma$ of $\zeta(s)$. We define the weighted sum

[TABLE]

and use the following explicit formula, proved in [7] (see also Thm. 28 of [10]).

Lemma 3**.**

For non-integer $x>0$ we have

[TABLE]

where

[TABLE]

The bound on $\epsilon(x)$ has been reduced from [7], as we can write

[TABLE]

and

[TABLE]

Using a linear combination of equation (5), we can examine the distribution of prime powers in the interval $(x,x+h)$. For $2\leq\Delta<\sqrt{x}\log x\leq h\leq x$, let

[TABLE]

This leads to the identity

[TABLE]

which can be verified by expanding both sides. Notice that over $x\leq n\leq x+h$, the sum on the LHS is equal to $\psi(x+h)-\psi(x)$. We thus aim to estimate this expression by bounding the RHS of (2.1). Using Lemma 3 in the above equation gives the following.

Lemma 4**.**

Let $2\leq\Delta<h\leq x$ with $x\notin\mathbb{Z}$. Then

[TABLE]

where

[TABLE]

and

[TABLE]

It remains to estimate the sum over zeros. We will split it into three sums,

[TABLE]

where $\alpha>0$ and $\beta>0$ are parameters we can later optimise over.

Lemma 5**.**

Let $2\leq\Delta<h\leq x$ and assume RH. We have

[TABLE]

provided that $\beta x/\Delta\geq\gamma_{1}=14.13\ldots$, the ordinate of the first zero of $\zeta(s)$.

Proof.

On RH, one has

[TABLE]

The result follows from Lemma 1(ii) of Skewes [15], that for all $T\geq\gamma_{1}$,

[TABLE]

∎

The following lemmas require estimates on the zero-counting function $N(T)$, which counts the number of zeros of $\zeta(s)$ in the critical strip $0<\beta<1$ with $0<\gamma\leq T$. Backlund [1] showed that $N(T)=P(T)+Q(T)$, where

[TABLE]

and $Q(T)=O(\log T)$. Hasanalizade, Shen, and Wong [8, Cor. 1.2] have given the most recent explicit version of this, of

[TABLE]

with $a_{1}=0.1038$, $a_{2}=0.2573$, and $a_{3}=9.3675$, for all $T\geq e$.

Lemma 6**.**

Let $2\leq\Delta<h\leq x$ and assume RH. We have

[TABLE]

Proof.

We can write

[TABLE]

so, under RH, one has

[TABLE]

With (8), we can use

[TABLE]

from which the result immediately follows. ∎

For the middle sum of (7), we will use the following lemma. It follows directly from Lemma 3 of [2], in whose notation we use $\phi(\gamma)=\gamma^{-1}$, and takes constants $A_{0}$ and $A_{1}$ from Trudgian [16, Thm. 2.2] and $A_{2}$ from [2, Lem. 2].

Lemma 7**.**

For $2\pi\leq T_{1}\leq T_{2}$ we have

[TABLE]

where $|Q(T)|\leq R(T)$, defined in (8), and

[TABLE]

with $A_{0}=2.067$, $A_{1}=0.059$, $A_{2}=1/150$.

Lemma 8**.**

Let $2\leq\Delta<h\leq x$ and assume RH. For $\alpha x/h\geq 15$ we have

[TABLE]

Proof.

We can write

[TABLE]

and so bounding trivially gives

[TABLE]

It follows that

[TABLE]

on which we apply Lemma 7, and bound the smaller order terms with the assumption of $T_{1}\geq 15$ to obtain the result. Note that the bound on $T_{1}$ is to reduce the constant $5.4$, but not restrict $\alpha$ too much. ∎

2.2 Bounding the PNT in intervals

From Lemma 4 we can write

[TABLE]

As the smooth weight has $|w(n)|\leq 1$, the above bound is no greater than

[TABLE]

The largest term in this bound comes from the sum over $\rho$, in particular, the section estimated in Lemma 8. Larger $\Delta$ results in a smaller main-term constant, so we will set $\Delta=C\sqrt{x}\log x$ and later choose an optimal value of $C\in(0,1)$. The reason for not taking larger $\Delta$ is two-fold: to keep $\Delta<h$ and ensure the smaller terms in (10) are $O(\sqrt{x}\log x)$.

To bound the sum over prime powers we can use Montgomery and Vaughan’s version of the Brun–Titchmarsh theorem for primes in intervals [12, Eq. 1.12]. Defining $\theta(x)=\sum_{p\leq x}\log p$, equation (1.12) of [12] implies

[TABLE]

The contribution from higher prime powers is relatively small, and can be bounded with explicit estimates on the difference between the Chebyshev functions $\psi(x)$ and $\theta(x)$. Costa Pereira [5, Thm. 2,4,5] gives lower bounds for different ranges of $x$. These can be combined into

[TABLE]

for all $x\geq 2187$. Broadbent et al. [3, Cor. 5.1] give

[TABLE]

with $\alpha_{1}=1+1.93378\cdot 10^{-8}$ and $\alpha_{2}=2.69$ for all $x\geq e^{10}$. Thus, we have

[TABLE]

where $E_{1}(x)=\alpha_{1}(x+h+\Delta)^{\frac{1}{2}}+\alpha_{2}(x+h+\Delta)^{\frac{1}{3}}-0.999(x+h)^{\frac{1}{2}}-\frac{2}{3}(x+h)^{\frac{1}{3}}$, and is bounded by $E_{1}(x)\leq\beta_{1}x^{\frac{1}{2}}+\beta_{2}x^{\frac{1}{3}}$ with

[TABLE]

Here and hereafter, let $x_{0}=e^{10}$. For $x\geq x_{0}$ we can bound the smaller order terms in (10),

[TABLE]

where, for $h\leq x^{t}$ with $t<1$,

[TABLE]

This, along with Lemmas 5 and 6, allow us to bound

[TABLE]

where

[TABLE]

For $\sqrt{x}\log x\leq h\leq x^{t}$ we have

[TABLE]

where, for $x\geq x_{0}\geq e^{\beta/C}$ and $0<\alpha\leq 5$, we can take

[TABLE]

The first term in (13) can be estimated with Lemma 8, so that

[TABLE]

in which, assuming $100e^{-10}\leq\frac{\alpha\beta}{4\pi^{2}C}\leq 100$, we can take

[TABLE]

Note that the assumption for $\alpha$ and $\beta$ is to ensure certain terms are bounded for all $x\geq x_{0}$. Combining estimates, we have

[TABLE]

where $K_{4}=K_{3}+K_{2}$. It remains to optimise over the parameters. Before deciding these values, recall that we have made the assumptions $\beta\leq 10C$,

[TABLE]

The restriction on $\alpha$ will be satisfied for all $\sqrt{x}\log x\leq h\leq x^{\frac{3}{4}}$ if we take $\alpha\geq 15x_{0}^{-\frac{1}{4}}$. Optimising over $C$, $\alpha$, and $\beta$ to minimise $K_{4}$, we find that choosing $C=0.25$ and $\alpha=\beta=1.35$ allows us to take $K_{4}=2$ for all $x\geq x_{0}$.