Riesz-type criteria for the Riemann hypothesis

TL;DR

This paper generalizes classical criteria for the Riemann hypothesis, establishing new Riesz-type bounds by extending Hardy and Littlewood's identity, thus providing alternative equivalent conditions for the hypothesis.

Contribution

It introduces a one-variable generalization of Hardy and Littlewood's identity and derives new Riesz-type criteria for the Riemann hypothesis.

Findings

Established a generalized identity linking to the Riemann hypothesis.

Derived bounds consistent with Riesz and Hardy-Littlewood criteria.

Provided new equivalent conditions 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 $x \rightarrow\infty$, 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.