Harmonic analysis and Number Theory

• Grant number: 13/14881-9
• Support Opportunities: Scholarships in Brazil - Doctorate
• Start date: November 01, 2013
• End date: March 31, 2017
• Field of knowledge: Physical Sciences and Mathematics - Mathematics
• Principal Investigator: Dimitar Kolev Dimitrov
• Grantee: Willian Diego Oliveira
• Host Institution: Instituto de Biociências, Letras e Ciências Exatas (IBILCE). Universidade Estadual Paulista (UNESP). Campus de São José do Rio Preto. São José do Rio Preto , SP, Brazil
• Associated research grant: 09/13832-9 - Orthogonal polynomials, special functions and applications, AP.TEM

Abstract

Among the efficient tools to study the distribution of the prime numbers, which is a principal theme in Number Theory, are the so-called Vinogradov-Korobov method and also the analysis of the entire functions, represented in terms of a Fourier transform, whose zeros are all real. Both tools have their origin in Harmonic Analysis. Other applications which reveal the fundamental role of this branch of Analysis in Number Theory can be encountered in the excellent book of one of the most celebrated contemporary experts in Analytic Number Theory, Hugh Montgomery, entitled "Ten Lectures on the Interface Between Analytic Number Theory and Harmonic Analysis".It is worth mentioning that two historic and brilliant results concerning old problems about prime numbers, obtained very recently, were based on the so called "seiving"' which isbased in the analysis of certain sums of trigonometric polynomials.On the other hand, recent research shows that the characterisation of the Fourier transforms with real zeros is directly related not only to the Riemann hypothesis but also to fundamental problems in Statistical Mechanics.The objective of this project is to study the applications of Harmonic Analysis in Analytic Number Theory. (AU)

Scientific publications

(References retrieved automatically from Web of Science and SciELO through information on FAPESP grants and their corresponding numbers as mentioned in the publications by the authors)

OLIVEIRA, WILLIAN D.. Zeros of Dirichlet polynomials via a density criterion. JOURNAL OF NUMBER THEORY, v. 203, p. 80-94, OCT 2019. (13/14881-9, 17/15223-6, 16/09906-0)

DIMITROV, DIMITAR K.; OLIVEIRA, WILLIAN D.. Sign regularity of Maclaurin coefficients of functions in the Laguerre-Polya class. JOURNAL D ANALYSE MATHEMATIQUE, v. 137, n. 2, p. 897-911, MAR 2019. (13/14881-9, 16/09906-0)

Academic Publications

(References retrieved automatically from State of São Paulo Research Institutions)

OLIVEIRA, Willian Diego. Zeros of Dirichlet series and of functions in the Laguerre-Pólya class. 2017. Doctoral Thesis - Universidade Estadual Paulista (Unesp).