## Explicit results for Euler’s factorial series in arithmetic progressions under GRH

### Volume 211 / 2023

#### Abstract

We study Euler’s factorial series $F_p(t)=\sum _{n=0}^\infty n!t^n$ in the $p$-adic domain under the Generalized Riemann Hypothesis. First, we show that if we consider primes in $k\varphi (m)/(k+1)$ residue classes in the reduced residue system modulo $m$, then under certain explicit extra conditions we must have $\lambda _0+\lambda _1F_p(\alpha _1)+\cdots +\lambda _kF_p(\alpha _k) \neq 0$ for at least one such prime. We also prove an explicit $p$-adic lower bound for that linear form. Secondly, we consider the case where we take primes in arithmetic progressions from more than $k\varphi (m)/(k+1)$ residue classes. Then there is an infinite collection of intervals each containing at least one prime which is in those arithmetic progressions and for which we have $\lambda _0+\lambda _1F_p(\alpha _1)+\cdots +\lambda _kF_p(\alpha _k) \neq 0$. We also derive an explicit $p$-adic lower bound for the linear form.