RIESZ-TYPE CRITERIA FOR THE RIEMANN HYPOTHESIS

In 1916, Riesz proved that the Riemann Hypothesis is equivalent to the bound sigma(infinity)(n=1) mu(n)/n(2 ) exp (- x/n(2)) = O-is an element of (x(-3/4+is an element of)), as x -> infinity, for any is an element of > 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.