Euler's divergent series in arithmetic progressions

TL;DR

This paper investigates the distribution of primes in certain residue classes related to Euler's factorial series, showing that infinitely many primes satisfy specific non-vanishing conditions in these classes.

Contribution

It establishes the existence of infinitely many primes in specific residue classes where Euler's factorial series does not vanish, extending understanding of prime distribution in relation to divergent series.

Findings

Infinitely many primes in certain residue classes satisfy non-vanishing conditions.

Distribution of primes linked to Euler's factorial series evaluated p-adically.

Results depend on the structure of residue classes and properties of Euler's series.

Abstract

Let $\xi$ and $m$ be integers satisfying $\xi\ne 0$ and $m\ge 3$. We show that for any given integers $a$ and $b$, $b \neq 0$, there are $\frac{\varphi(m)}{2}$ reduced residue classes modulo $m$ each containing infinitely many primes $p$ such that $a-bF_p(\xi) \ne 0$, where $F_p(\xi)=\sum_{n=0}^\infty n!\xi^n$ is the $p$-adic evaluation of Euler's factorial series at the point $\xi$.