A survey on Beurling-Selberg majorants and some consequences of the Riemann hypothesis

Abstract

This article briefly describes the recent advances on the Beurling-Selberg extremal problem in harmonic analysis and its connection with the theory of the Riemann zeta-function. In particular, under the Riemann hypothesis, this extremal tool provides improved bounds for the size of ζ(s) in the critical strip, for the argument function S(t) and for its antiderivative, the function S1(t).