On the behavior of some two-variable \(p\)-adic \(L\)-functions.
type: Item
instance of: scholarly article
title: On the behavior of some two-variable \(p\)-adic \(L\)-functions.
zbMATH Open document ID: 1067.11076
DOI: 10.1016/S0022-314X(02)00031-8
author: Paul Thomas Young
published in: Journal of Number Theory
publication date: 2003-04-28T00:00:00Z
review text: The Kubota--Leopoldt \(p\)-adic \(L\)-function \(L_p(s,\chi)\) possesses a natural extension to a two variable function \(L_p(s,t,\chi)\) introduced by \textit{G.\ Fox} [Enseign. Math., II. Sér. 46, No. 3--4, 225--278 (2000; Zbl 0999.11073)]. This function is defined for all \(t\) in the valuation ring of \({\mathbb C}_p\) and satisfies \(L_p(s,0,\chi)=L_p(s,\chi)\). The author expresses \(L_p(s,t,\chi)\) by means of a \(p\)-adic integral. For \(\chi\) a power of the Teichmüller character, this representation allows him to extend the domain of definition of \(L_p(s,t,\chi)\) to include \(t\in q^{-1}{\mathbb Z}_p\) (here, as usual, \(q\) equals \(p\) or 4 according as \(p>2\) or \(p=2\), respectively). As another consequence the author obtains new systems of congruences for generalized Bernoulli polynomials improving previous results by Fox, H. S. Gunaratne, and himself. If \(f_\chi\), the conductor of \(\chi\), is not a \(p\)-power, the results also yield a generalization of formulas for the derivative \(L_p'(0,\chi)\). Finally, for \(f_\chi\) unequal to 1 or \(qp^e\) (\(e\geq 1\)), the author constructs two different extensions of \(L_2(s,t,\chi)\) to values of \(t\) in \(2^{-1}{\mathbb Z}_2\).
Mathematics Subject Classification ID: 11S40
Mathematics Subject Classification ID: 11B68
Mathematics Subject Classification ID: 11R42
zbMATH DE Number: 1902868
zbMATH Keywords: \(p\)-adic \(L\)-functions
zbMATH Keywords: Bernoulli polynomials
MaRDI profile type: Publication
cites work: Some Congruences for the Bernoulli Numbers
cites work: Arithmetic Properties of Generalized Bernoulli Numbers.
cites work: The p-Adic Log Gamma Function and p-Adic Euler Constants
cites work: On the values of p-adic L-functions at positive integers
cites work: Q3967633
cites work: On the behavior of \(p\)-adic \(L\)-functions at \(s=0\)
cites work: A \(p\)-adic \(L\)-function of two variables
cites work: Q4842417
cites work: Lectures on P-Adic L-Functions. (AM-74)
cites work: Q5579667
cites work: Q3943047
cites work: Congruences for Bernoulli, Euler, and Stirling numbers
cites work: Kummer congruences for values of Bernoulli and Euler polynomials
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description: scientific article; zbMATH DE number 1902868