Ernvall-Hytönen, A-M., Matala-aho, T. & Seppälä, L. (2019) Euler’s divergent series in arithmetic progressions. Journal of integer sequences 22(2), article 19.2.2. https://cs.uwaterloo.ca/journals/JIS/VOL22/Seppala/seppala2.pdf

### Euler’s divergent series in arithmetic progressions

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Author: Ernvall-Hytönen, Anne-Maria1; Matala-aho, Tapani2; Seppälä, Louna2
Organizations: 1Matematik och Statistik, Åbo Akademi University, Domkyrkotorget 1, 20500 Åbo, Finland
2Matematiikka, PL 8000, 90014 Oulun yliopisto, Finland
Format: article
Version: published version
Access: open
Online Access: PDF Full Text (PDF, 0.1 MB)
Language: English
Published: University of Waterloo, 2019
Publish Date: 2020-04-07
Description:

# Abstract

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 $$ξ$$.

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Series: Journal of integer sequences
ISSN: 1530-7638
ISSN-E: 1530-7638
Volume: 22
Issue: 2
Article number: 19.2.2
Type of Publication: A1 Journal article – refereed
Field of Science: 111 Mathematics
Subjects: