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arXiv · 2202.00637

Riesz-type criteria for the Riemann hypothesis

Archit Agarwal·Meghali Garg·Bibekananda Maji
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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.

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Archit Agarwal, Meghali Garg, Bibekananda Maji. 2022-01-31. Riesz-type criteria for the Riemann hypothesis. https://arxiv.org/abs/2202.00637

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