Equivalent criteria for the Riemann hypothesis for a general class of $L$-functions
| Autor: | Garg, Meghali, Maji, Bibekananda |
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| Rok vydání: | 2024 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | 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. Comment: 30 pages, 2 tables |
| Databáze: | arXiv |
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