A polylogarithmic measure associated with a path on $\Pbb ^1\setminus \{ 0,1,\infty \}$ and a $P$-adic Hurwitz zeta function

TL;DR

This paper introduces a new elementary measure linked to paths on the punctured projective line, which enables the construction of p-adic Hurwitz zeta functions through integration, connecting geometric paths with special functions.

Contribution

It defines a novel measure associated with paths on the punctured projective line and demonstrates its use in constructing p-adic Hurwitz zeta functions in a simple, elementary manner.

Findings

The measure can be defined very elementary.

Integration against this measure yields p-adic Hurwitz zeta functions.

The approach connects geometric paths with special functions in p-adic analysis.

Abstract

With every path on a projective line minus zero, one and infinity there is associated a measure. We are studying a sum of two such measures associated to paths from the tangential point at zero to roots of one. We show that the obtained measure can be defined very elementary. Integrating agaist this measure we get p-adic Hurwitz zeta functions constructed previously by Shiratani.