University of Oulu

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.

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)
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Language: English
Published: University of Waterloo, 2019
Publish Date: 2020-04-07


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
Copyright information: © 2019 The Authors.