Reference : Prime divisors of the l-Genocchi numbers and the ubiquity of Ramanujan-style congruen...
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Physical, chemical, mathematical & earth Sciences : Mathematics
Prime divisors of the l-Genocchi numbers and the ubiquity of Ramanujan-style congruences of level l
Moree, Pieter mailto [Max Planck Institute Mathematics Bonn]
Sgobba, Pietro mailto [University of Luxembourg > Faculty of Science, Technology and Medicine (FSTM) > Department of Mathematics (DMATH) >]
[en] l-Genocchi numbers ; l-regularity ; Ramanujan type congruences ; Artin's primitive root conjecture
[en] 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,..., 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.

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