Euler's factorial series at algebraic integer points

JOURNAL OF NUMBER THEORY

卷:206

Article

Seppala, Louna¹ 

[1] Univ Oulu, Res Unit Math Sci, POB 8000, Oulu 90014, Finland

关键词: Diophantine approximation;    Divergent series;    Number field;    Padc approximation;    p-Adic;    Valuation;   

DOI  :  10.1016/j.jnt.2019.06.013

来源: Elsevier

【 摘 要 】

We study a linear form in the values of Euler's series F(t) = Sigma(infinity)(n=0) n!t(n) at algebraic integer points alpha(j) is an element of Z(K,) j = 1,..., m, belonging to a number field K. In the two main results it is shown that there exists a non-Archimedean valuation v vertical bar p of the field K such that the linear form Lambda(v) = lambda(0) + lambda F-1(v) (alpha(1)) +...+ lambda F-m(v) (alpha(m)), lambda(i) is an element of Z(K), does not vanish. The second result contains a lower bound for the v-adic absolute value of 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 Pade approximations to the generalised factorial series Sigma(infinity)(n=0) (Pi(n-1)(k=0) P(k)) t(n), where P(x) is a polynomial of degree one. (C) 2019 Elsevier Inc. All rights reserved.

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