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.