Zeros of linear combinations of Dirichlet L-functions on the critical line

Speaker: Jérémy Dousselin

Date: Mon, Mar 25, 2024

Location: PIMS, University of British Columbia, Zoom

Conference: Analytic Aspects of L-functions and Applications to Number Theory

Subject: Mathematics, Number Theory

Class: Scientific

CRG: L-Functions in Analytic Number Theory

Abstract:

Fix N1 and let L1,L2,,LN be Dirichlet L-functions with distinct, primitive and even Dirichlet characters. We assume that these functions satisfy the same functional equation. Let F(s)∶=c1L1(s)+c2L2(s)++cNLN(s) be a linear combination of these functions (cjR are distinct). F is known to have two kinds of zeros: trivial ones, and non-trivial zeros which are confined in a vertical strip. We denote the number of non-trivial zeros ρ with F(ρ)\leq TbyN(T),andweletN_\theta(T)bethenumberofthesezerosthatareonthecriticalline.Attheendofthe90s,Selbergprovedthatthislinearcombinationhadapositiveproportionofzerosonthecriticalline,byshowingthat\kappa F∶=\lim \inf T (N_\theta(2T)−N_\theta(T))/(N(2T)−N(T))\geq c/N^2forsomec>0.Ourgoalistoprovideanexplicitvalueforc,andalsotoimprovethelowerboundabovebyshowingthat\kappa F \geq 2.16\times 10^{-6}/(N \log N),foranylargeenoughN$.