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| Hlavní autoři: | , |
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| Médium: | Preprint |
| Vydáno: |
2024
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| Témata: | |
| On-line přístup: | https://arxiv.org/abs/2409.17708 |
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| _version_ | 1866929519390097408 |
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| author | Garg, Meghali Maji, Bibekananda |
| author_facet | Garg, Meghali Maji, Bibekananda |
| contents | 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{μ(n)}{n} \exp\left({-\frac{x}{n^2}}\right) = O_ε\left( x^{-\frac{1}{4}+ ε} \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 $ζ(s)$. Furthermore, we generalize an identity of Ramanujan, Hardy and Littlewood for Chandrasekharan-Narasimhan class of $L$-functions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_17708 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Equivalent criteria for the Riemann hypothesis for a general class of $L$-functions Garg, Meghali Maji, Bibekananda Number Theory Primary 11M06, Secondary 11M26 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{μ(n)}{n} \exp\left({-\frac{x}{n^2}}\right) = O_ε\left( x^{-\frac{1}{4}+ ε} \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 $ζ(s)$. Furthermore, we generalize an identity of Ramanujan, Hardy and Littlewood for Chandrasekharan-Narasimhan class of $L$-functions. |
| title | Equivalent criteria for the Riemann hypothesis for a general class of $L$-functions |
| topic | Number Theory Primary 11M06, Secondary 11M26 |
| url | https://arxiv.org/abs/2409.17708 |