Let β be any fixed prime number. We define the β-Genocchi numbers
by Gnβ:=β(1ββn)Bnβ, with Bnβ 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 β-Genocchi irregular if it divides
at least one of the β-Genocchi numbers G2β,G4β,β¦,Gpβ3β, and
β-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 β-Genocchi irregular primes in a prescribed arithmetic progression in
case β is odd. The case β=2 was already dealt with by Hu, Kim, Moree
and Sha (2019).
Using similar methods we study the prime factors of (1ββn)B2nβ/2n and
(1+βn)B2nβ/2n. This allows us to estimate the number of primes pβ€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
β.Comment: 27 page