$p$-adic Congruences of Generalized Euler Numbers and Relations to Even Zeta Values

Abstract

Generalized Euler numbers have previously been studied mainly from a combinatorial viewpoint. The purpose of this paper is to explore them from $p$-adic and analytic perspectives. To this end, we introduce congruential Euler numbers, a new family extending generalized Euler numbers. We establish several $p$-adic congruences for these numbers, including an answer to a conjecture related to Lehmer numbers. Furthermore, using complex analytic methods, we derive expressions for even zeta values in terms of congruential Euler numbers. These results suggest that certain types of these numbers may possess connections with arithmetic or analytic structures beyond their purely combinatorial role as generalizations of Euler-type numbers.