Riesz-type criteria for the Riemann hypothesis

In 1916, Riesz proved that the Riemann Hypothesis is equivalent to the bound ∑n=1∞μ(n)n2exp⁡(−xn2)=Oϵ(x−34+ϵ)\sum _{n=1}^\infty \frac {\mu (n)}{n^2} \exp \left ( - \frac {x}{n^2} \right ) = O_{\epsilon } \left ( x^{-\frac {3}{4} + \epsilon } \right ), as x→∞x \rightarrow \infty, for any ϵ>0\epsilon >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.