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

Let 8 be any fixed prime number. We define the 8-Genocchi numbers by Gn := 8(1 - 8n)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 8-Genocchi irregular if it divides at least one of the 8-Genocchi numbers G2, G4, ... , Gp-3, and 8 -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 8-Genocchi irregular primes in a prescribed arithmetic progression in case 8 is odd. The case 8 = 2 was already dealt with by Hu et al. (2019) [14]. Using similar methods we study the prime factors of (1-8n)B2n/2n and (1 + 8n)B2n/2n. This allows us to estimate the number of primes p & LE; 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 8.& COPY; 2023 Elsevier Inc. All rights reserved.