# 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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