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Journal of the Ramanujan Mathematical Society

Volume 37, Issue 4, December 2022  pp. 301–318.

A polylogarithmic measure associated with a path on P{1}{0,1,∞} and a P-adic Hurwitz zeta function

Authors:  Zdzisław Wojtkowiak
Author institution: Université de Nice-Sophia Antipolis, Département de Mathématiques, Laboratoire Jean Alexandre Dieudonné, U.R.A. au C.N.R.S., N◦ 168, Parc Valrose – B.P. N◦ 71 06108 Nice Cedex 2, France and 179 Piste de l’Uesti, 06910 Pierrefeu, France

Summary:  With every path on P{1}{Q} {0,1,∞} there is associated a measure on Z{p}. The group Z{p}{×} acts on measures. We consider two measures. One measure is associated to a path from {01} to a root of unity ξ of order prime to p. Another measure is associated to a path from {01} to ξ{−1} and next it is acted by −1 ∈ Z{p}{×}. We show that the sum of these measures can be defined in a very elementary way. Integrating against this sum of measures we get p-adic Hurwitz zeta functions constructed previously by Shiratani.

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