Prime divisors of ℓ-Genocchi numbers and the ubiquity of Ramanujan-style congruences of level ℓ

Abstract

Let ℓ be any fixed prime number. We define the ℓ-Genocchi numbers by Gn:=ℓ(1−ℓn)Bn, with Bn 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 ℓ-Genocchi irregular if it divides at least one of the ℓ-Genocchi numbers G2,G4,…,Gp−3, and ℓ-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 ℓ-Genocchi irregular primes in a prescribed arithmetic progression in case ℓ is odd. The case ℓ=2 was already dealt with by Hu et al. (2019) [14]. Using similar methods we study the prime factors of (1−ℓn)B2n/2n and (1+ℓn)B2n/2n. This allows us to estimate the number of primes p≤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 ℓ.