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

Fix $\delta \in (0,1]$, ${\sigma }_0\in [0,1)$ and a real-valued function $\epsilon (x)$ for which ${\mathrm{lim\hspace{0.17em}sup}}_{x\to \mathrm{\infty }}\epsilon (x)\le 0$. For every set of primes $\mathcal{P}$ whose counting function ${\pi }_{\mathcal{P}}(x)$ satisfies an estimate of the form $${\pi }_{\mathcal{P}}(x)=\delta \pi (x)+O\left(x^{{\sigma }_0+\epsilon (x)}\right),$$ we define a zeta function ${\zeta }_{\mathcal{P}}(s)$ that is closely related to the Riemann zeta function $\zeta (s)$. For ${\sigma }_0\le \frac{1}{2}$, 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 $${\pi }_{\mathcal{P}}(x)=\delta \pi (x)+O((\log \log x{)}^{\epsilon (x)}),$$ 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 $$\{2,547,1229,1993,2749,3581,4421,5281\dots \},$$ which consists of $2$ and every hundredth prime thereafter.