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Topics in Fourier analysis and number theory


The aim of this project is to discuss problems related to Fourier analysis and Number Theory, with a special emphasis on the behavior of the Riemann zeta function. The principal interest of the author of the project are in topics related to zeros of polynomials and entire functions and especially in zeros of orthogonal polynomials and special functions. Some recent results of ours provide necessary and sufficient conditions that a Fourier transform of a Borel measure possesses only real zeros. It is worth mentioning that the problem to obtain such conditions was formulated by Pólya who himself was motivated by his efforts in settle the Riemann hypothesis by merely classical methods. Because of it, recently we have approximated our interests to the origin of the problem. We study fundamental in Analytic Number Theory. The PhD student Wiliann Oliveira, supported by a FAPESP fellowship, in his MSc thesis presented properties of the Riemann zeta function and a proof of the Prime Number Theorem. Recently we work on theoretical questions, studying properties of entire functions, defined as Fourier transforms, that simmulate the properties of the so-called Riemann $\xi$ function. We would like to have a better vision on the properties of the Riemann $\zeta$ e $\xi$ functions in order to approach some questions in Analytic Number Theory. The visiting researcher, Professor Aleksandar Ivic is one of the most celebrated contemporary experts in Number Theory and the author of the most cited book on Riemann's zeta function. We shall have intensive discussions and activities during his eventual visit in order that we learn a lot about the zeta function form the best possible source. (AU)

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