Euler's divergent series in arithmetic progressions

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 $\xi$.