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: embargoed
Language: English
Published: Elsevier, 2020
Publish Date: 2021-07-18
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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