The Kronecker limit formulas via the distribution relation

TL;DR

This paper provides a new proof of the classical Kronecker limit formulas using the distribution relation of Eisenstein-Kronecker series and extends these results to $p$-adic analogues for $p$-adic Eisenstein-Kronecker functions.

Contribution

It introduces a novel proof technique for the Kronecker limit formulas and establishes their $p$-adic analogues, expanding the understanding of these functions.

Findings

Classical Kronecker limit formulas proved using distribution relations.

Established $p$-adic analogues of the Kronecker limit formulas.

Extended the theory of Eisenstein-Kronecker functions to the $p$-adic setting.

Abstract

In this paper, we give a proof of the classical Kronecker limit formulas using the distribution relation of the Eisenstein-Kronecker series. Using a similar idea, we then prove $p$-adic analogues of the Kronecker limit formulas for the $p$-adic Eisenstein-Kronecker functions defined in our previous paper.