Sum Expressions for Kubota-Leopoldt $p$-adic $L$-functions

TL;DR

This paper derives sum expressions for Kubota-Leopoldt $p$-adic $L$-functions for all odd primes, removing previous restrictions, and applies these to reprove known formulas and explore convergence and algebraic relations.

Contribution

It establishes general sum expressions for $p$-adic $L$-functions without restrictions on $p$ or Euler factors, expanding previous results.

Findings

Derived sum expressions valid for all odd primes

Reproved Ferrero-Greenberg formula using new methods

Discussed convergence and relations to Stickelberger elements

Abstract

When $p$ is an odd prime, Delbourgo observed that any Kubota-Leopoldt $p$-adic $L$-function, when multiplied by an auxiliary Euler factor, can be written as an infinite sum. We shall establish such expressions without restriction on $p$, and without the Euler factor when the character is nontrivial, by computing the periods of appropriate measures. As an application, we will reprove the Ferrero-Greenberg formula for the derivative $L_p'(0,\chi)$. We will also discuss the convergence of sum expressions in terms of elementary $p$-adic analysis, as well as their relation to Stickelberger elements; such discussions in turn give alternative proofs of the validity of sum expressions.