Critical Zeros and Unconditional Mean Value Theorems for twisted $\hbox{PGL}(2)$ and $\hbox{PGL}(3)$ $\mathrm{L}$-functions

Abstract

Let $Π_{0}$ be a cuspidal automorphic representation of $\mathrm{PGL}_{3}(\mathbb{A}_{\mathbb{Q}})$. In this paper, we use Levinson&#39;s method to prove that, as $Q\to \infty$, at least $1/9$ of the zeros of the $L$-functions $L(s, Π_{0}\,\times\, χ)$ lie on the critical line, where $χ$ ranges over the family of primitive Dirichlet characters of conductor up to $Q$. This result is unconditional when $Π_{0}$ is self-dual, and otherwise holds under a mild condition. The key technical input is a new asymptotic formula with a power-saving error term for the mean square of the product of $L(s, Π_{0}\times χ)$ and a Dirichlet polynomial with arbitrary coefficients in both the $T$- and $Q$-aspects for the range $Q^ε\le T \le Q^{1/3-ε}$. When $T=Q^ε$, our asymptotic formula allows Dirichlet polynomials of length $θ<1/2-ε$; when $θ=0$, it gives a strong error term of size $O_ε(Q^{7/4+ε})$. Furthermore, our result provides evidence for the CFKRS conjectures for large twists and large vertical shifts. We also obtain corresponding results for $\mathrm{PGL}_{2}(\mathbb{A}_{\mathbb{Q}})$, which are fully unconditional, quantitatively stronger, and also appear to be new. This work develops a refined, flexible, and uniform version of the Asymptotic Large Sieve for $L$-functions that does not require any unproven progress toward the Generalized Ramanujan Conjecture. The arithmetic of $Π_{0}$ plays a crucial and delicate role in our argument. This work also makes extensive use of Mathematica to handle various elaborate Hecke algebra computations. Our mean value theorem is readily applicable to many other problems in analytic number theory.