Prime divisors of $\ell$-Genocchi numbers and the ubiquity of Ramanujan-style congruences of level $\ell$

Abstract

Let $\ell$ be any fixed prime number. We define the $\ell$-Genocchi numbers by $G_n:=\ell(1-\ell^n)B_n$, with $B_n$ the $n$-th Bernoulli number. They are integers. We introduce and study a variant of Kummer's notion of regularity of primes. We say that an odd prime $p$ is $\ell$-Genocchi irregular if it divides at least one of the $\ell$-Genocchi numbers $G_2,G_4,\ldots, G_{p-3}$, and $\ell$-regular otherwise. With the help of techniques used in the study of Artin's primitive root conjecture, we give asymptotic estimates for the number of $\ell$-Genocchi irregular primes in a prescribed arithmetic progression in case $\ell$ is odd. The case $\ell=2$ was already dealt with by Hu, Kim, Moree and Sha (2019). Using similar methods we study the prime factors of $(1-\ell^n)B_{2n}/2n$ and $(1+\ell^n)B_{2n}/2n$. This allows us to estimate the number of primes $p\leq x$ for which there exist modulo $p$ Ramanujan-style congruences between the Fourier coefficients of an Eisenstein series and some cusp form of prime level $\ell$.