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Riesz-type criteria for the Riemann hypothesis

In 1916, Riesz proved that the Riemann hypothesis is equivalent to the bound $\sum_{n=1}^\infty \frac{μ(n)}{n^2} \exp\left( - \frac{x}{n^2} \right) = O_ε \left( x^{-\frac{3}{4} + ε} \right)$, as $x \rightarrow\infty$, for any $ε>0$. Around the same time, Hardy and Littlewood gave another equivalent criteria 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.