Artin meets Bernoulli: prime divisors of generalized Genocchi numbers

Contact: Pieter Moree (moree@mpim-bonn.mpg.de)

Let $\ell$ be any fixed prime number. We study the prime factors of the generalized Genocchi numbers $G_n:=\ell(1-\ell^n)B_n$ and some related numbers, with $B_n$ the $n$-th Bernoulli number. With the help of techniques used in the study of Artin's primitive root conjecture, we consider the distribution of their prime factors in a prescribed arithmetic progression. 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 $\ell$. For odd $\ell$ this is joint work with Pietro Sgobba. The case $\ell=2$ is the classical Genocchi number setting, where there is a connection with class number divisibility. I investigated this with Hu, Kim and Sha (2019).