Irregular primes with respect to Genocchi numbers and Artin's primitive root conjecture

Abstract

We introduce and study a variant of Kummer's notion of (ir)regularity of primes which we call G-(ir)regularity and is based on Genocchi rather than Bernoulli numbers. We say that an odd prime p is G-irregular if it divides at least one of the Genocchi numbers G2,G4,…,Gp−3, and G-regular otherwise. We show that, as in Kummer's case, G-irregularity is related to the divisibility of some class number. Furthermore, we obtain some results on the distribution of G-irregular primes. In particular, we show that each primitive residue class contains infinitely many G-irregular primes and establish non-trivial lower bounds for their number up to a given bound x as x tends to infinity. As a byproduct, we obtain some results on the distribution of primes in arithmetic progressions with a prescribed near-primitive root.