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

Fix $δ\in(0,1]$, $σ_0\in[0,1)$ 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 $π_{\mathcal P}(x)$ satisfies an estimate of the form $$π_{\mathcal P}(x)=δ\,π(x)+O\bigl(x^{σ_0+\varepsilon(x)}\bigr),$$ we define a zeta function $ζ_{\mathcal P}(s)$ that is closely related to the Riemann zeta function $ζ(s)$. For $σ_0\le\frac12$, we show that the Riemann hypothesis is equivalent to the non-vanishing of $ζ_{\mathcal P}(s)$ in the region $\{σ>\frac12\}$. For every set of primes ${\mathcal P}$ that contains the prime $2$ and whose counting function satisfies an estimate of the form $$π_{\mathcal P}(x)=δ\,π(x)+O\bigl((\log\log x)^{\varepsilon(x)}\bigr),$$ 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.