In 1916, Riesz proved that the Riemann Hypothesis is equivalent to the bound ∑ n = 1 ∞ μ ( n ) n 2 exp ⁡ ( − x n 2 ) = O ϵ ( x − 3 4 + 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.

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