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Harmonic analysis and Number Theory

Grant number: 13/14881-9
Support type:Scholarships in Brazil - Doctorate
Effective date (Start): November 01, 2013
Effective date (End): March 31, 2017
Field of knowledge:Physical Sciences and Mathematics - Mathematics
Principal Investigator:Dimitar Kolev Dimitrov
Grantee:Willian Diego Oliveira
Home 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


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. Web of Science Citations: 2.
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. Web of Science Citations: 0.
Academic Publications
(References retrieved automatically from State of São Paulo Research Institutions)
OLIVEIRA, Willian Diego. Zeros de séries de Dirichlet e de funções na classe de Laguerre-Pólya. 2017. 87 f. Doctoral Thesis - Universidade Estadual Paulista "Júlio de Mesquita Filho" Instituto de Biociências, Letras e Ciências Exatas..

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