Topics in additive and multiplicative number theory.

Seminar in Mathematics

Abstract: In this talk we will focus on innovative approaches in additive and multiplicative number theory and the interplay between additive and multiplicative techniques. In particular, we shall discuss past and recent work as well as future directions in three different topics:
- The analysis of asymptotic formulae for mixed moments of the Riemann-zeta function. Such examinations are performed both unconditionally and under the assumption of a weaker version of the abc conjecture to sharpen the corresponding error terms. Analogous questions in the context of Dirichlet L-functions have been recently considered by Fouvry-Kowalski-Michel.
-Refinements on Vu’s thin basis theorem in Waring’s problem. These are achieved incorporating both innovations in the application of the probabilistic method and recent developments of Brudern-Wooley concerning the use of the circle method in Waring’s problem.
-Asymptotic evaluations of divisor correlations in almost all short intervals, which involves utilising both the circle method and Matomaki-Radziwill machinery. This includes upcoming joint work with Yu-Chen Sun and Mengdi Wang.