Euler’s divergent series in arithmetic progressions

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