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p-Adic Eigen-Eunctions for Kubert Distributions

Published online by Cambridge University Press:  20 November 2018

Neal Koblitz
Neal Koblitz*
Affiliation:
University of Washington, Seattle, Washington
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Functions on R (or on R/Z, or Q/Z, or the interval (0,1)) which satisfy the identity

1.1

for positive integers m and fixed complex s, appear in several branches of mathematics (see [8], p. 65-68). They have recently been studied systematically by Kubert [6] and Milnor [12]. Milnor showed that for each complex s there is a one-dimensional space of even functions and a one-dimensional space of odd functions which satisfy (1.1). These functions can be expressed in terms of either the Hurwitz partial zeta-function or the polylogarithm functions.

My purpose is to prove an analogous theorem for p-adic functions. The p-adic analog is slightly more general; it allows for a Dirichlet character χ0(m) in front of ms–l in (1.1). The functions satisfying (1.1) turn out to be p-adic “partial Dirichlet L-functions”, functions of two p-adic variables (x, s) and one character variable χ0 which specialize to partial zeta-functions when χ0 is trivial and to Kubota-Leopoldt L-functions when x = 0.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 35 , Issue 4 , 01 August 1983 , pp. 674 - 686
Copyright
Copyright © Canadian Mathematical Society 1983

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