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 $\varlimsup _{x\rightarrow \infty }\varepsilon (x)\leqslant 0$. For every set of primes $\mathcal {P}$ whose counting function $\pi _\mathcal {P}(x)$ satisfies an estimate of the form

$$\begin{aligned} \pi _\mathcal {P}(x)=\delta \,\pi (x)+O\bigl (x^{\sigma _0+\varepsilon (x)}\bigr ), \end{aligned}$$

we define a zeta function $\zeta _\mathcal {P}(s)$ that is closely related to the Riemann zeta function $\zeta (s)$. For $\sigma _0\leqslant \frac{1}{2}$, we show that the Riemann hypothesis is equivalent to the non-vanishing of $\zeta _\mathcal {P}(s)$ in the region $\{\sigma >\frac{1}{2}\}$.

For every set of primes $\mathcal {P}$ that contains the prime 2 and whose counting function satisfies an estimate of the form

$$\begin{aligned} \pi _\mathcal {P}(x)=\delta \,\pi (x)+O\bigl ((\log \log x)^{\varepsilon (x)}\bigr ), \end{aligned}$$

we show that $\mathcal {P}$ is an exact 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 exact asymptotic additive basis for $\mathbb {N}$ is provided by the set

$$\begin{aligned} \{2,547,1229,1993,2749,3581,4421,5281\ldots \}, \end{aligned}$$

which consists of 2 and every hundredth prime thereafter.