A note on arithmetic congruences

TL;DR

This paper investigates arithmetic congruences related to p-adic L-functions, connecting classical number theory results with modern p-adic analysis, and proving a new congruence of Ankeny--Artin--Chowla type.

Contribution

It introduces a novel congruence for coefficients of p-adic L-functions associated with real quadratic fields, linking classical and modern number theory.

Findings

Proves a new Ankeny--Artin--Chowla-type congruence for prime power moduli.

Shows classical Bernoulli and Wilson congruences fit into p-adic L-function theory.

Connects p-adic L-functions with classical number theoretic congruences.

Abstract

By analyzing the coefficients of the power series defining the Kubota--Leopoldt $p$-adic $L$-function associated to the non-trivial character of a real quadratic field, we prove a congruence of Ankeny--Artin--Chowla-type for prime power modulus. Additionally, we show how some classical congruences relating Bernoulli numbers and Wilson quotients fit naturally into the theory of the $p$-adic Riemann zeta function.