  In 1916, Riesz proved that the Riemann hypothesis is equivalent to the bound
$\sum_{n=1}^\infty \frac{\mu(n)}{n^2} \exp\left( - \frac{x}{n^2} \right) =
O_{\epsilon} \left( x^{-\frac{3}{4} + \epsilon} \right)$, as $x
\rightarrow\infty$, for any $\epsilon >0$. 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.
