Riesz-type criteria for the Riemann hypothesis

Abstract

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 $x\to \infty$, 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.

14 pages, comments are welcome!