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

The $\theta =\mathrm{\infty }$ 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 $\theta =\mathrm{\infty }$ conjecture for a family of $L$-functions implies a quasi-Riemann Hypothesis for that family.