Reference : Prime divisors of the l-Genocchi numbers and the ubiquity of Ramanujan-style congruen...
E-prints/Working papers : First made available on ORBilu
Physical, chemical, mathematical & earth Sciences : Mathematics
http://hdl.handle.net/10993/51204
Prime divisors of the l-Genocchi numbers and the ubiquity of Ramanujan-style congruences of level l
English
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) >]
Undated
No
[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.
Researchers
http://hdl.handle.net/10993/51204
https://arxiv.org/abs/2209.08047

File(s) associated to this reference

Fulltext file(s):

FileCommentaryVersionSizeAccess
Open access
MoreeSgobba_arxiv_1.pdfAuthor preprint469.86 kBView/Open

Bookmark and Share SFX Query

All documents in ORBilu are protected by a user license.