A criterion related to the Riemann Hypothesis

TL;DR

This paper explores a specific distance measure related to the Riemann Hypothesis within the Nyman-Beurling-Báez-Duarte framework, demonstrating how the presence of zeros off the critical line affects this criterion.

Contribution

It introduces a new criterion based on the distance measure that indicates the non-validity of the Riemann Hypothesis if zeros off the critical line exist.

Findings

Shows how zeros off the critical line influence the distance measure

Provides a criterion for disproving the Riemann Hypothesis

Connects zero distribution to functional approximation in the Nyman-Beurling approach

Abstract

A crucial role in the Nyman-Beurling-B\'aez-Duarte approach to the Riemann Hypothesis is played by the distance \[ d_N^2:=\inf_{A_N}\frac{1}{2\pi}\int_{-\infty}^\infty\left|1-\zeta A_N\left(\frac{1}{2}+it\right)\right|^2\frac{dt}{\frac{1}{4}+t^2}\:, \] where the infimum is over all Dirichlet polynomials $$A_N(s)=\sum_{n=1}^{N}\frac{a_n}{n^s}$$ of length $N$. In this paper we investigate $d_N^2$ under the assumption that the Riemann zeta function has four non-trivial zeros off the critical line. Thus we obtain a criterion for the non validity of the Riemann Hypothesis.

Equations

$$d_N^2:=\underset{A_N}{\inf }\frac{1}{2\pi }{\int }_{-\infty }^{\infty }{\left|1-\zeta A_N\left(\frac{1}{2}+it\right)\right|}^2\frac{dt}{\frac{1}{4}+t^2},$$

$$A_N(s)=\sum _{n=1}^N\frac{a_n}{n^s}$$

$$\underset{N\to \infty }{\lim }{d}_N^2=0,$$

$$d_N^2:=\underset{A_N}{\inf }\frac{1}{2\pi }{\int }_{-\infty }^{\infty }{\left|1-\zeta A_N\left(\frac{1}{2}+it\right)\right|}^2\frac{dt}{\frac{1}{4}+t^2}$$

$$A_N(s)=\sum _{n=1}^N\frac{a_n}{n^s}$$

$$\underset{N\to \infty }{\mathrm{liminf}}{d}_N^2\log N\ge \sum _{Re(\rho )=\frac{1}{2}}\frac{m(\rho {)}^2}{|\rho {|}^2},$$

$$d_N^2\sim \frac{1}{\log N}\sum _{Re(\rho )=\frac{1}{2}}\frac{m(\rho {)}^2}{|\rho {|}^2}.$$

$$\sum _{Re(\rho )=\frac{1}{2}}\frac{m(\rho )}{|\rho {|}^2}=2+\gamma -\log 4\pi ,$$

$$V_N(s):=\sum _{n=1}^N\left(1-\frac{\log n}{\log N}\right)\frac{\mu (n)}{n^s}.$$

$$\sum _{|Im(\rho )|\le T}\frac{1}{|{\zeta }^{\prime }(\rho ){|}^2}\ll T^{\frac{3}{2}-\delta }$$

$$\frac{1}{2\pi }{\int }_{-\infty }^{\infty }{\left|1-\zeta V_N\left(\frac{1}{2}+it\right)\right|}^2\frac{dt}{\frac{1}{4}+t^2}\sim \frac{2+\gamma -\log 4\pi }{\log N}.$$

$$\Lambda (s)=\Lambda (1-s),$$

$$\Lambda (s):={\pi }^{-s/2}\Gamma \left(\frac{s}{2}\right)\zeta (s)$$

$$\zeta (\sigma +i\gamma )=\zeta (1-\sigma +i\gamma )=\zeta (\sigma -i\gamma )=\zeta (1-\sigma -i\gamma )=0.$$

$$\sum _{|Im(\rho )|\le T}\frac{1}{|{\zeta }^{\prime }(\rho ){|}^2}\ll T^{\frac{3}{2}-\delta }(T\to \infty ),$$

$$\frac{1}{2\pi }{\int }_{-\infty }^{\infty }{\left|1-\zeta V_N\left(\frac{1}{2}+it\right)\right|}^2\frac{dt}{\frac{1}{4}+t^2}$$

$$=\frac{1}{{\log }^{2}N}\left(A{N}^{2{\sigma }_0-1}\cos (2{\gamma }_0\log N)+B{N}^{2{\sigma }_0-1}\right)\left(1+O(N^{\frac{1}{2}-{\sigma }_0+ϵ})\right),$$

$$\zeta (\sigma +it)\ll |t{|}^{3ϵ}(|t|\to \infty ),$$

$$\frac{1}{2}-ϵ\le \sigma \le \frac{1}{2}+ϵ.$$

$$\mathcal{G}:=\left\{\sigma +it:\sigma >\frac{1}{2}\right\}\setminus \left\{\left[\frac{1}{2},1\right]\cup \left[\frac{1}{2}+i{\gamma }_0,{\sigma }_0+i{\gamma }_0\right]\cup \left[\frac{1}{2}-i{\gamma }_0,{\sigma }_0-i{\gamma }_0\right]\right\}.$$

$$Re(\log \zeta (z))=\log |\zeta (z)|<A\log t$$

$$|\log \zeta (z)|\le \frac{3-2\delta }{\frac{1}{2}\delta }A\log t+\frac{3-\frac{3}{2}\delta }{\frac{1}{2}\delta }|\log |\zeta (2+it)||<A{\delta }^{-1}\log t.$$

$$\zeta (z)=O(t^{ϵ}),\text{for every σ>21,}$$

$$\left|\zeta \left(\frac{1}{2}-ϵ+it\right)\right|=O\left(|t{|}^{3ϵ}\right).$$

$$R_N(\rho ,s):=Re{s}_{z=\rho }\frac{N^{z-s}}{\zeta (z)(z-s{)}^2}$$

$$F_s(z):=\pi z^s\sum _{n=1}^{\infty }(-1{)}^n\frac{(2\pi {)}^{2n+1}{z}^{2n}}{(2n)!\zeta (2n+1)(2n+s{)}^2}$$

$$V_N(s)=\frac{1}{\zeta (s)}\left(1-\frac{1}{\log N}\frac{{\zeta }^{\prime }}{\zeta }(s)\right)+\frac{1}{\log N}\sum _{\rho }{R}_N(\rho ,s)+\frac{1}{\log N}{F}_s\left(\frac{1}{N}\right),$$

$$\sum _{\rho ,Re(\rho )=\frac{1}{2}}{R}_N(\rho ,s)\ll N^{\mp ϵ}|s{|}^{\frac{3}{4}-\frac{\delta }{2}+ϵ}.$$

$$N^{\pm ϵ}\sum _{\rho ,|\rho |=\frac{1}{2}}{R}_N(\rho ,s)\ll \sum _{\begin{array}{c}|\rho -s|<\frac{|\rho |}{2}\\ Re(\rho )=\frac{1}{2}\end{array}}\frac{1}{|{\zeta }^{\prime }(\rho )||\rho -s{|}^2}+1.$$

$${\Sigma }^{(1)}(N,s):=\frac{1}{\log N}\sum _{\rho :Re(\rho )=\frac{1}{2}}{R}_N(\rho ,s)$$

Full text

[TABLE]

where the infimum is over all Dirichlet polynomials

[TABLE]

of length $N$.

In this paper we investigate $d_{N}^{2}$ under the assumption that the Riemann zeta function has four non-trivial zeros off the critical line. Thus we obtain a criterion for the non validity of the Riemann Hypothesis.

Key words: Riemann hypothesis, Riemann zeta function, Nyman-Beurling-Báez-Duarte criterion.

2000 Mathematics Subject Classification: 30C15, 11M26

1. Introduction

The Nyman-Beurling-Báez-Duarte approach to the Riemann hypothesis asserts that the Riemann hypothesis is true, if and only if

[TABLE]

where

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and the infimum is over all Dirichlet polynomials

[TABLE]

of length $N$ (see [3]).

Burnol [4], improving on work of Báez-Duarte, Balazard, Landreau and Saias [1], [2] showed that

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where $m(\rho)$ denotes the multiplicity of the zero $\rho$.

This lower bound is believed to be optimal and one expects that

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Under the Riemann hypothesis one has

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where $\gamma$ is the Euler-Mascheroni constant.

S. Bettin, J. B. Conrey and D. W. Farmer [3] prove (*) under an additional assumption and also identify the Dirichlet polynomials $A_{N}$, for which the expected infimum in (1.1) is assumed. They prove (Theorem 1 of [3]):

Let

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If the Riemann hypothesis is true and if

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for some $\delta>0$, then

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In this paper we investigate the expression (1.3) under an assumption contrary to the Riemann hypothesis: There are exactly four nontrivial zeros off the critical line. We observe that nontrivial zeros off the critical line always appear as quadruplets. Indeed, if $\zeta(\rho)=0$ for $\rho=\sigma+i\gamma$ with $1>\sigma>\frac{1}{2}$, $\gamma>0$, then from the functional equation

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where

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and the trivial relation $\zeta(\bar{s})=\overline{\zeta(s)}$, we obtain that

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We prove the following:

Theorem 1.1**.**

Let $\sigma_{0}>1/2$, $\gamma_{0}>0$, $\zeta(\sigma_{0}\pm i\gamma_{0})=\zeta(1-\sigma_{0}\pm i\gamma_{0})=0$ and $\zeta(\sigma+i\gamma)\neq 0$ for all other $\sigma+i\gamma$ with $\sigma>1/2$. Assume that

[TABLE]

for some $\delta>0$. Then, there are constants $A=A(\sigma_{0},\gamma_{0})$ and $B=B(\sigma_{0},\gamma_{0})\in\mathbb{R}$, such that for all $\epsilon>0$:

[TABLE]

[TABLE]

for some $\epsilon>0$.

2. Preliminary Lemmas and Definitions

Lemma 2.1**.**

Let $\epsilon>0$ be fixed but arbitrarily small. Under the assumptions of Theorem 1.1 we have

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for

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Proof.

The estimate (2.1) is well known as the Lindelöf hypothesis, which is a consequence of the Riemann hypothesis. In [5], the Lindelöf hypothesis is proven on the assumption of the Riemann hypothesis. This proof may be adapted to the new situation by slight modifications.

The function $\log\zeta(s)$ is holomorphic in the domain

[TABLE]

Let now $\frac{1}{2}<\sigma^{*}\leq\sigma\leq 1$. As in [5], let $z=\sigma+it$, but now $|t|$ sufficiently large.

We apply the Borel-Carathéodory theorem to the function $\log\zeta(z)$ and the circles with centre $2+it$ and radius $\frac{3}{2}-\frac{1}{2}\delta$ and $\frac{3}{2}-\delta$, ($0<\delta<\frac{1}{2}$).

On the larger circle

[TABLE]

for a fixed positive constant $A$. Hence on the smaller circle

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We now apply Hadamard’s three circle theorem as in [5]. The proof there can be taken over without change, to obtain

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which is (14.2.5) of [5].

By the functional equation (1.4) we obtain

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The claim (2.1) now follows from (2.2), (2.3) and the theorem of Phragmén-Lindelöf. ∎

Definition 2.2**.**

For $\rho$ a non-trivial zero of $\zeta(s)$ let

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and

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Lemma 2.3**.**

If $0<Re(s)<1$, then

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where the sum is over all distinct non-trivial zeros of $\zeta(s)$.

Proof.

This is Lemma 2 of [3]. ∎

Lemma 2.4**.**

Let $\epsilon>0$. Under the assumptions of Theorem 1.1 we have

[TABLE]

Proof.

The proof is identical to the proof of Lemma 3 of [3]. There the summation condition $Re(s)=1/2$ is not needed, since the Riemann hypothesis is assumed. ∎

Lemma 2.5**.**

[TABLE]

Proof.

This is (5) of [3]. ∎

Definition 2.6**.**

We set

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and

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3. Proof of Theorem 1.1

We closely follow the proof of Theorem 1 in [3]. We have

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By Lemma 2.3 and Definition 2.6 this is

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We now expand the product in (3.1) and separately estimate the products that do not contain terms $\Sigma^{(2)}$ and the products consisting of a term $\Sigma^{(2)}$ and another term.

The asymptotic finally is obtained by asymptotically evaluating the products consisting only of factors $\Sigma^{(2)}$.

We closely follow [3]. It follows from Lemmas 2.1, 2.3, 2.4 that

[TABLE]

Now by Lemma 2.4 and the trivial estimate

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all the other terms in (3.1) not containing factors $\Sigma^{(2)}$ are trivially $O(1/\log^{2}N)$ apart from

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where we set

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and use the bound

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which follows from Lemma 2.1 by the well-known estimate for the derivatives of a holomorphic function. By moving the line of integration to $Re(s)=\frac{1}{2}+\epsilon$, we get that the contribution from the products not containing $\Sigma^{(2)}$ is

[TABLE]

We now come to the products that contain the factor $\Sigma^{(2)}(N,s)$.

They may be handled by adding the factor $N^{\sigma_{0}-\frac{1}{2}+\epsilon}$ stemming from $N^{z-s}$ in Definition 2.2.

These estimates yield the error-term in Theorem 1.1. The main term finally is obtained by evaluating the contribution with two factors $\Sigma^{(2)}$ and by observing that the integral in (1.3) is real.

∎