Zeta functions and asymptotic additive bases with some unusual sets of primes

TL;DR

This paper explores the properties of zeta functions associated with special prime sets, linking their behavior to the Riemann hypothesis and demonstrating that certain prime sets form asymptotic additive bases for natural numbers.

Contribution

It introduces a new class of zeta functions related to prime sets with specific counting function estimates and establishes their connection to the Riemann hypothesis and additive basis properties.

Findings

Non-vanishing of the zeta functions is equivalent to the Riemann hypothesis for certain prime sets.

Prime sets with specific counting function estimates form asymptotic additive bases for natural numbers.

An explicit example of such a prime set is provided, containing 2 and every hundredth prime thereafter.

Abstract

Fix $\delta\in(0,1]$, $\sigma_0\in[0,1)$ and a real-valued function $\varepsilon(x)$ for which $\limsup_{x\to\infty}\varepsilon(x)\le 0$. For every set of primes ${\mathcal P}$ whose counting function $\pi_{\mathcal P}(x)$ satisfies an estimate of the form $$\pi_{\mathcal P}(x)=\delta\,\pi(x)+O\bigl(x^{\sigma_0+\varepsilon(x)}\bigr),$$ we define a zeta function $\zeta_{\mathcal P}(s)$ that is closely related to the Riemann zeta function $\zeta(s)$. For $\sigma_0\le\frac12$, we show that the Riemann hypothesis is equivalent to the non-vanishing of $\zeta_{\mathcal P}(s)$ in the region $\{\sigma>\frac12\}$. For every set of primes ${\mathcal P}$ that contains the prime $2$ and whose counting function satisfies an estimate of the form $$\pi_{\mathcal P}(x)=\delta\,\pi(x)+O\bigl((\log\log x)^{\varepsilon(x)}\bigr),$$ we show that ${\mathcal P}$ is an asymptotic additive basis for ${\mathbb N}$, i.e., for some integer $h=h({\mathcal P})>0$ the sumset $h{\mathcal P}$ contains all but finitely many natural numbers. For example, an asymptotic additive basis for ${\mathbb N}$ is provided by the set $$ \{2,547,1229,1993,2749,3581,4421,5281\ldots\}, $$ which consists of $2$ and every hundredth prime thereafter.