2022/09/16 by Pieter Moree, Moree, Pieter, Pietro Sgobba +1
Mathematics · #11B68 #Advanced Algebra and Geometry #Advanced Mathematical Identities #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT) #Primary 11A07 #secondary 11F33
paper · pdf · doi:10.48550/arxiv.2209.08047
openalex publication_date 2022/09/16 · openalex created_date 2022/09/20 · openalex updated_date 2026/07/28
Let ℓ be any fixed prime number. We define the ℓ-Genocchi numbers by Gn:=ℓ(1-ℓn)Bn, with Bn 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 ℓ-Genocchi irregular if it divides at least one of the ℓ-Genocchi numbers G2,G4,…, Gp-3, and ℓ-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 ℓ-Genocchi irregular primes in a prescribed arithmetic progression in case ℓ is odd. The case ℓ=2 was already dealt with by Hu, Kim, Moree and Sha (2019). Using similar methods we study the prime factors of (1-ℓn)B2n/2n and (1+ℓn)B2n/2n. This allows us to estimate the number of primes p≤ 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 ℓ.