Louna Seppälä, Euler’s factorial series at algebraic integer points, Journal of Number Theory, Volume 206, 2020, Pages 250-281, ISSN 0022-314X, https://doi.org/10.1016/j.jnt.2019.06.013

### Euler‘s factorial series at algebraic integer points

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Author: Seppälä, Louna1
Organizations: 1Research Unit of Mathematical Sciences, University of Oulu, P.O. Box 8000, 90014, Finland
Format: article
Version: accepted version
Access: open
Online Access: PDF Full Text (PDF, 0.3 MB)
Persistent link: http://urn.fi/urn:nbn:fi-fe202003309642
Language: English
Published: Elsevier, 2020
Publish Date: 2020-03-30
Description:

# Abstract

We study a linear form in the values of Euler’s series $$F(t)=\sum\nolimits_{n=0}^\infty n!t^n$$ at algebraic integer points $$α_j∈\mathbb{Z}_\mathbb{K}, j=1,…,m$$, belonging to a number field $$\mathbb{K}$$. In the two main results it is shown that there exists a non-Archimedean valuation $$v\vert p$$ of the field $$\mathbb{K}$$ such that the linear form $${\mathrm\Lambda}_v=\lambda_0+\lambda_1F_v(\alpha_1)+\dots+\lambda_mF_v(\alpha_m)$$, $$\lambda_i\in{\mathbb{Z}}_\mathbb{K}$$, does not vanish. The second result contains a lower bound for the v-adic absolute value of $${\mathrm\Lambda}_v$$, and the first one is also extended to the case of primes in residue classes. On the way to the main results, we present explicit Padé approximations to the generalised factorial series $$\sum\nolimits_{n=0}^\infty{\left(\prod\nolimits_{k=0}^{n-1}P(k)\right)}t^n$$, where $$P(x)$$ is a polynomial of degree one.

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Series: Journal of number theory
ISSN: 0022-314X
ISSN-E: 1096-1658
ISSN-L: 0022-314X
Volume: 206
Pages: 250 - 281
DOI: 10.1016/j.jnt.2019.06.013
OADOI: https://oadoi.org/10.1016/j.jnt.2019.06.013
Type of Publication: A1 Journal article – refereed
Field of Science: 111 Mathematics
Subjects:
Funding: The work of the author was supported by the University of Oulu Scholarship Foundation and the Vilho, Yrjö and Kalle Väisälä Foundation of the Finnish Academy of Science and Letters.
Copyright information: © 2019 Elsevier Inc. This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/.
https://creativecommons.org/licenses/by-nc-nd/4.0/