RIESZ-TYPE CRITERIA FOR THE RIEMANN HYPOTHESIS

Abstract

In 1916, Riesz proved that the Riemann Hypothesis is equivalent to the bound Formula Presented, for any ∊ > 0. Around the same time, Hardy and Littlewood gave another equivalent criterion for the Riemann Hypothesis while correcting an identity of Ramanujan. In the present paper, we establish a one-variable generalization of the identity of Hardy and Littlewood and as an application, we provide Riesz-type criteria for the Riemann Hypothesis. In particular, we obtain the bound given by Riesz as well as the bound of Hardy and Littlewood. © 2022 American Mathematical Society.