Journal of Integer Sequences, Vol. 22 (2019), Article 19.2.2

Euler's Divergent Series in Arithmetic Progressions


Anne-Maria Ernvall-Hytönen
Matematik och Statistik
Åbo Akademi University
Domkyrkotorget 1
20500 Åbo
Finland

Tapani Matala-aho and Louna Seppälä
Matematiikka
PL 8000
90014 Oulun yliopisto
Finland

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 $\xi$.

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Received November 7 2018; revised version received January 14 2019. Published in Journal of Integer Sequences, February 22 2019.

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