The $θ= \infty$ Conjecture and the Riemann Hypothesis for Automorphic $L$-functions
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
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2026
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| _version_ | 1866910250946265088 |
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| author | Dong, Anji Wattanawanichkul, Nawapan Zaharescu, Alexandru |
| author_facet | Dong, Anji Wattanawanichkul, Nawapan Zaharescu, Alexandru |
| contents | The $θ=\infty$ conjecture asserts that the mollified second moments of the Riemann zeta function remain bounded for mollifiers of arbitrary polynomial length. We investigate an analogue of this conjecture for automorphic $L$-functions associated with cuspidal representations of $\text{GL}_m(\mathbb{A}_{\mathbb{Q}})$, exploring its implications for the distribution of their nontrivial zeros. Extending the framework of Bettin and Gonek, we prove that if the mollified second moments of these $L$-functions remain suitably bounded for mollifiers of arbitrary polynomial length, then the $L$-functions are non-vanishing in corresponding regions of the critical strip. Furthermore, we establish a version of this criterion for families of $L$-functions, demonstrating that the $θ= \infty$ conjecture for a family of $L$-functions implies a quasi-Riemann Hypothesis for that family. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_24363 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The $θ= \infty$ Conjecture and the Riemann Hypothesis for Automorphic $L$-functions Dong, Anji Wattanawanichkul, Nawapan Zaharescu, Alexandru Number Theory Primary: 11F66, Secondary: 11M06, 11M26 The $θ=\infty$ conjecture asserts that the mollified second moments of the Riemann zeta function remain bounded for mollifiers of arbitrary polynomial length. We investigate an analogue of this conjecture for automorphic $L$-functions associated with cuspidal representations of $\text{GL}_m(\mathbb{A}_{\mathbb{Q}})$, exploring its implications for the distribution of their nontrivial zeros. Extending the framework of Bettin and Gonek, we prove that if the mollified second moments of these $L$-functions remain suitably bounded for mollifiers of arbitrary polynomial length, then the $L$-functions are non-vanishing in corresponding regions of the critical strip. Furthermore, we establish a version of this criterion for families of $L$-functions, demonstrating that the $θ= \infty$ conjecture for a family of $L$-functions implies a quasi-Riemann Hypothesis for that family. |
| title | The $θ= \infty$ Conjecture and the Riemann Hypothesis for Automorphic $L$-functions |
| topic | Number Theory Primary: 11F66, Secondary: 11M06, 11M26 |
| url | https://arxiv.org/abs/2605.24363 |