Zeros of partial sums of L-functions

Abstract

We consider a certain class of multiplicative functions f : N → C. Let F (s) = ∞ n=1 f (n)n −s be the associated Dirichlet series and FN (s) = n≤N f (n)n −s be the truncated Dirichlet series. In this setting, we obtain new Halász-type results for the logarithmic mean value of f . More precisely, we prove estimates for the sum x n=1 f (n)/n in terms of the size of |F (1 + 1/ log x)| and show that these estimates are sharp. As a consequence of our mean value estimates, we establish non-trivial zero-free regions for these partial sums FN (s).In particular, we study the zero distribution of partial sums of the Dedekind zeta function of a number field K. More precisely, we give some improved results for the number of zeros upto height T as well as new zero density results for the number of zeros up to height T , lying to the right of Re(s) = σ, where σ ≥ 1/2.

Zeros of Dirichlet polynomials

“…We note here that 10) of [25]). Making use of this bound, we will estimate the expression in ( 23) by considering two cases.…”

“…Since the zero-free region (9) follows from Theorem 2.3 of [25], we devote this section to proving the former part of Theorem 1.1.…”

“…Their method is general enough to be applied to completely multiplicative functions with values inside the unit disc, but fails for the more general class of multiplicative functions. In [25], the authors bridged this gap by obtaining a zero-free region for partial sums associated to functions in a suitable subclass C k of multiplicative functions. Let k be a positive real number.…”

“…This class of multiplicative functions includes most multiplicative functions studied in the literature and has been an object of active interest recently, for instance in the work of Granville, Harper and Soundararajan [6], as well as Drappeau, Granville and Shao [8,7,5]. In [25,Theorem 2.3], the authors proved that if f ∈ C k , then the corresponding Dirichlet polynomial F N (s) does not vanish in the half plane…”

Une note sur la densité des zéros des sommes partielles de la fonction zeta de Dedekind sur un corps quadratique

“…Ce résultat fut généralisé dans [6] pour ( ) dans le cas où est premier puis pour un quelconque dans [13]. L'étude des zéros des sommes partielles représente toujours un angle d'attaque de la conjecture de Riemann voir, par exemple, [8] et sert à mieux comprendre la répartition des zéros des fonctions L de Dirichlet de façon plus générale voir, par exemple, [16,17,18].…”