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- W4296412307 abstract "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 $G_2,G_4,ldots, G_{p-3}$, 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 $pleq x$ for which there exist modulo $p$ Ramanujan-style congruences between the Fourier coefficients of an Eisenstein series and some cusp form of prime level $ell$." @default.
- W4296412307 created "2022-09-20" @default.
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- W4296412307 date "2022-09-16" @default.
- W4296412307 modified "2023-09-24" @default.
- W4296412307 title "Prime divisors of $ell$-Genocchi numbers and the ubiquity of Ramanujan-style congruences of level $ell$" @default.
- W4296412307 doi "https://doi.org/10.48550/arxiv.2209.08047" @default.
- W4296412307 hasPublicationYear "2022" @default.
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