$p$-adic Congruencens of Generalized Euler Numbers and Relations to Even Zeta Value

TL;DR

This paper introduces congruential Euler numbers, establishes their $p$-adic congruences, and connects them to even zeta values through complex analysis, advancing understanding of number theory and special functions.

Contribution

It generalizes Euler numbers to congruential Euler numbers, proves their $p$-adic congruences, and links them to even zeta values, addressing a conjecture on Lehmer numbers.

Findings

Proved $p$-adic congruences for congruential Euler numbers.

Provided expressions of even zeta values via these numbers.

Answered a conjecture related to Lehmer numbers.

Abstract

In this article, we introduce congruential Euler numbers, which are a further generalization of generalized Euler numbers. We prove the $p$-adic congruences of congruential Euler numbers, which include answers to a conjecture related to Lehmer numbers. We also provide expressions of even zeta values using congruential Euler numbers via complex analysis.