An explicit density estimate for Dirichlet $L$-series

Abstract

Dirichlet L-series L(s, χ) = ∑ n≥1 χ(n)n −s associated to primitive Dirichlet characters χ are one of the keys to the distribution of primes. Even the simple case χ = 1 which corresponds to the Riemann zeta-function contains many informations on primes and on the Farey dissection. There have been many generalizations of these notions, and they all have arithmetical properties and/or applications, see [45, 29, 33] for instance. Investigations concerning these functions range over many directions, see [14] or [43]. We note furthermore that Dirichlet characters have been the subject of numerous studies, see [2, 50, 4]; Dirichlet series in themselves are still mysterious, see [3] and [6]. One of the main problem concerns the location of the zeroes of these functions in the strip 0 < <s < 1; the Generalized Riemann Hypothesis asserts that all of those are on the line <s = 1/2. We concentrate in this paper on estimating