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arXiv · 2605.24363

The $\theta = \infty$ Conjecture and the Riemann Hypothesis for Automorphic $L$-functions

Anji Dong·Nawapan Wattanawanichkul·Alexandru Zaharescu
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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.

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Anji Dong, Nawapan Wattanawanichkul, Alexandru Zaharescu. 2026-05-23. The $\theta = \infty$ Conjecture and the Riemann Hypothesis for Automorphic $L$-functions. https://arxiv.org/abs/2605.24363

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