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 $b$, $b\ne 0$, there are $\frac{\phi (m)}{2}$ reduced residue classes modulo $m$ each containing infinitely many primes $p$ such that $a-b{F}_p(\xi )\ne 0$, where $F_p(\xi )={\sum }_{n=0}^{\infty }n!{\xi }^n$ is the $p$-adic evaluation of Euler's factorial series at the point $\xi$.