Equivalent criteria for the Riemann hypothesis for a general class of $L$-functions

TL;DR

This paper establishes new equivalent criteria for the Riemann hypothesis across a broad class of $L$-functions, including cusp forms and the Riemann zeta function, extending classical bounds and identities.

Contribution

It proves the conjecture linking bounds to RH for general $L$-functions and introduces novel criteria and identities for these functions.

Findings

Established equivalent criteria for RH for cusp form $L$-functions.

Derived a new form of RH criterion for the Riemann zeta function.

Generalized Ramanujan, Hardy, and Littlewood identities for Chandrasekharan-Narasimhan class.

Abstract

In 1916, Riesz gave an equivalent criterion for the Riemann hypothesis (RH). Inspired from Riesz's criterion, Hardy and Littlewood showed that RH is equivalent to the following bound: \begin{align*} P_1(x):= \sum_{n=1}^\infty \frac{\mu(n)}{n} \exp\left({-\frac{x}{n^2}}\right) = O_{\epsilon}\left( x^{-\frac{1}{4}+ \epsilon } \right), \quad \mathrm{as}\,\, x \rightarrow \infty. \end{align*} Recently, the authors extended the above bound for the generalized Riemann hypothesis for Dirichlet $L$-functions and gave a conjecture for a class of ``nice'' $L$-functions. In this paper, we settle this conjecture. In particular, we give equivalent criteria for the Riemann hypothesis for $L$-functions associated to cusp forms. We also obtain an entirely novel form of equivalent criteria for the Riemann hypothesis of $\zeta(s)$. Furthermore, we generalize an identity of Ramanujan, Hardy and Littlewood for Chandrasekharan-Narasimhan class of $L$-functions.