Abstract:The $\theta=\infty$ 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 $\theta = \infty$ conjecture for a family of $L$-functions implies a quasi-Riemann Hypothesis for that family.
| Comments: | 11 pages |
| Subjects: | Number Theory (math.NT) |
| MSC classes: | Primary: 11F66, Secondary: 11M06, 11M26 |
| Cite as: | arXiv:2605.24363 [math.NT] |
| (or arXiv:2605.24363v1 [math.NT] for this version) | |
| https://doi.org/10.48550/arXiv.2605.24363 arXiv-issued DOI via DataCite (pending registration) |
From: Anji Dong [view email]
[v1]
Sat, 23 May 2026 02:59:37 UTC (15 KB)
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