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

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.