Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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
$\varepsilon(x)$ 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.

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The categories of this paper are: hep-th
25
The categories of this paper are: patt-sol
9
The categories of this paper are: nlin
32
The categories of this paper are: atom-ph
22
The categories of this paper are: chao-dyn
5
The categories of this paper are: gr-qc
3
The categories of this paper are: physics
33
The categories of this paper are: plasm-ph
8
The categories of this paper are: hep-ex
10
The categories of this paper are: q-bio
17
The categories of this paper are: q-fin
0
The categories of this paper are: hep-ph
24
The categories of this paper are: dg-ga
19
The categories of this paper are: eess
18
The categories of this paper are: econ
31
The categories of this paper are: ao-sci