Zeros of Dirichlet series and of functions in the Laguerre-Pólya class

Author(s)

Willian Diego Oliveira

• Document type: Doctoral Thesis
• Institution: Universidade Estadual Paulista (Unesp)
• Defense date: 2017-05-11
• Advisor: Dimitar Kolev Dimitrov

Abstract

We study topics related to zeros of Dirichlet series and entire functions. A large part of the thesis is devoted to the location of zeros of Dirichlet series via density criteria. We establish the Nyman-Beruling criterion for a wide class of Dirichlet series and the Báez-Duarte criterion for Dirichlet L-functions in the semi-plane R(s)>1/p, for p ∈ (1,2], as well as for zeros of Dirichlet polynomials in any semi-plane R(s)>r. An analog for the case of Dirichlet polynomials of a result of Burnol which is closely related to Báez-Duarte’s one is also established. A principal tool in the proof of the latter result is the solution of a natural extremal problem for Dirichlet polynomials inspired by Báez-Duarte’s result. We prove that the signs of the Maclaurin coefficients of a wide class of entire functions that belong to the Laguerre-Pólya class posses a regular behavior. (AU)

• FAPESP's process: 13/14881-9 - Harmonic analysis and Number Theory
• Grantee: Willian Diego Oliveira
• Support Opportunities: Scholarships in Brazil - Doctorate