On $p$-adic Euler $L$-functions

TL;DR

This paper introduces p-adic Euler L-functions via fermionic p-adic integrals, compares their values at negative integers with existing definitions, and explores their behavior at positive integers, extending classical number theory results.

Contribution

It defines p-adic Euler L-functions using fermionic integrals and establishes their properties, connecting with classical definitions and extending known results to Euler numbers.

Findings

Equivalence with previous definitions for odd conductor characters

Values at negative integers match classical p-adic L-functions

Behavior at positive integers parallels Bernoulli number results

Abstract

In this paper, we define the p-adic Euler L-functions using the fermionic p-adic integral on Zp. By computing the values of the p-adic Euler L-functions at negative integers, we show that for Dirichlet characters with odd conductor, this definition is quivalent to the previous definition following Kubata-Leopoldt and Washington's approach. We also study the behavior of p-adic Euler L-functions at positive integers. An interesting thing is that most of the results in Section 11.3.3 of Cohen's book [H. Cohen, Number Theory Vol. II: Analytic and Modern Tools, Graduate Texts in Mathematics, 240. Springer, New York, 2007] are also established if we replace the generalized Bernoulli numbers with the generalized Euler numbers.