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

Abstract: Let $Π<em>{0}$ be a cuspidal automorphic representation of $\mathrm{PGL}</em>{3}(\mathbb{A}<em>{\mathbb{Q}})$. In this paper, we use Levinson's method to prove that, as $Q\to \infty$, at least $1/9$ of the zeros of the $L$-functions $L(s, Π</em>{0}\,\times\, χ)$ 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 $Π<em>{0}$ 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 $L(s, Π</em>{0}\times χ)$ and a Dirichlet polynomial with arbitrary coefficients in both the $T$- and $Q$-aspects for the range $Q^ε\le T \le Q<sup>{1/3-ε}$. When $T=Q<sup>ε$, our asymptotic formula allows Dirichlet polynomials of length $θ&lt;1/2-ε$; when $θ=0$, it gives a strong error term of size $O_ε(Q<sup>{7/4+ε})$. Furthermore, our result provides evidence for the CFKRS conjectures for large twists and large vertical shifts. We also obtain corresponding results for $\mathrm{PGL}<em>{2}(\mathbb{A}</em>{\mathbb{Q}})$, 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.

• The paper establishes unconditional lower bounds on critical zeros, proving 1/3 and 1/9 proportions for PGL(2) and PGL(3) twisted L-functions respectively.
• It refines the asymptotic large sieve technique, achieving near-optimal Dirichlet polynomial lengths and power-saving error terms without reliance on the Generalized Ramanujan Conjecture.
• The work leverages combinatorial Hecke algebra and divisor-switching methods to manage complex arithmetic structures and derive uniform mean square formulas across spectral and modulus aspects.

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

Overview

This paper addresses both the horizontal and vertical mean value theory of automorphic $L$-functions, focusing on critical zeros and unconditional mean square formulae for twisted families of $L$-functions attached to automorphic representations of $\mathrm{PGL}(2)$ and $\mathrm{PGL}(3)$. The methods refine the asymptotic large sieve (ALS) framework, remove reliance on the Generalized Ramanujan Conjecture (GRC), and enable explicit lower bounds on the proportion of zeros on the critical line for families of twists of fixed automorphic forms.

Main Results and Numerical Proportions

The central result asserts that for a fixed cuspidal automorphic representation $\Pi_0$ of $\mathrm{PGL}_3(\mathbb{Q})$, as the modulus bound $Q\to\infty$, at least $\mathrm{PGL}(3)$0 of the zeros of $\mathrm{PGL}(3)$1 lie on the critical line, where $\mathrm{PGL}(3)$2 ranges over primitive Dirichlet characters of conductor up to $\mathrm{PGL}(3)$3. This result is unconditional for self-dual $\mathrm{PGL}(3)$4, and holds under a comparably mild $\mathrm{PGL}(3)$5-norm restriction on Dirichlet coefficients for the non-self-dual case. For $\mathrm{PGL}(3)$6, a parallel and entirely unconditional lower bound of $\mathrm{PGL}(3)$7 is proven.

The numerical proportions ($\mathrm{PGL}(3)$8 for $\mathrm{PGL}(3)$9, $L$0 for $L$1) match those predicted by straightforward mollification with Levinson's original mollifier. The result for $L$2 constitutes the first unconditional positive proportion of zeros on the critical line for such a high-degree family and is new for $L$3 in the general automorphic setting.

Mean Value Theorems and Asymptotic Formulae

At the heart of these results are new, uniform asymptotic formulae for the mean square of $L$4-functions twisted by Dirichlet polynomials (of arbitrary coefficients), simultaneously in the modulus ($L$5) and spectral ($L$6) aspects, over families of Dirichlet characters. The main theorem achieves a power-saving error term in the range $L$7 for $L$8, and $L$9 for $L$0. The permitted Dirichlet polynomial length is near-optimal for short intervals. For $L$1, mollifiers of length $L$2 are allowed, with the untwisted ($L$3) moment achieving error terms of $L$4.

Compared to previous ALS-based results, these mean value theorems:

• Hold with arbitrary coefficients, without requiring the $L$5 bound for automorphic coefficients—circumventing the impasse presented by present boundaries on GRC.
• Uniformly control off-diagonal and conductor-reducing terms via careful arithmetic, avoiding deep spectral estimates (such as shifted convolution) by leveraging the averaging over a sufficiently rich family.
• Make extensive use of combinatorial Hecke algebra calculations, managed via symbolic computation, to illuminate the arithmetic structure underlying divisor switches and twisted convolution.

Techniques: Asymptotic Large Sieve Without GRC

A major technical innovation is a new, flexible instantiation of the ALS that operates independently of any progress towards GRC. This is made possible by exploiting bilinear estimates for arithmetic GCD matrices and disentangling auxiliary coprimality conditions through repeated M\"obius inversion, ensuring the separation of variables necessary for effective use of the hybrid large sieve (in the sense of Gallagher).

Further, the use of divisor-switching, inspired by ideas from Gallagher and Conrey-Iwaniec-Soundararajan, is refined to manage auxiliary summation variables and address the delicate arithmetic interaction between mollifiers, twists, and spectral shifts. The symbolic computation with Mathematica provides exact formulae for the required local Euler products and their analytic properties.

Simple Zeros and Off-critical Line Zeros

Beyond mere counting of critical zeros, the methodology extends to simple zeros—proving a lower bound (albeit much smaller, e.g., $L$6 for $L$7) for the proportion of simple zeros on the critical line within these families. Moreover, the vertical (in $L$8) range can be scaled: for $L$9, positivity is achieved for critical line zeros up to height $L$0 for any $L$1.

Contrasts and Advances over Prior Work

Prior work either relied, tacitly or explicitly, on strong coefficient bounds or handled only "pure" Dirichlet $L$2-functions where such bounds come for free. The present paper achieves, for the first time:

• Unconditional positive-proportion results for higher rank automorphic $L$3-functions in twist families.
• ALS arguments that are entirely arithmetic, making them robust for future advances (e.g., handling Eisenstein series, non-self-dual primes, or more general underlying groups).

The technical apparatus to manage arithmetic complications—especially the interaction of divisor switching, auxiliary coprimality, and double Dirichlet series—establishes a toolkit for further unconditional mean value results in analytic number theory.

Theoretical and Practical Implications

From a theoretical perspective, this research offers new evidence for Random Matrix Theory-based conjectures about zero distributions in automorphic $L$4-function families, especially those predicted by CFKRS heuristics in the context of large twist or vertical shift limits. It also clarifies the limits of current unconditional methods, signaling where progress in shifted convolution or automorphic spectral theory would yield next improvements (e.g., breaking the $L$5 barrier for $L$6).

Practically, the developed mean value theorems are readily applicable to problems such as nonvanishing, simultaneous large value detection, and discrete moment computation for families of automorphic $L$7-functions. The flexibility in the Dirichlet coefficient configuration allows for adaptation to various mollifier-based zero-detection problems in the analytic theory of automorphic forms.

Directions for Future Research

Future work could include:

• Extending these methods to higher rank groups ($L$8 with $L$9), potentially in the context of the Rankin–Selberg convolution or products of $\mathrm{PGL}(2)$0-functions.
• Investigating analogous asymptotic formulae and critical zero bounds for families twisted by automorphic forms on other number fields or for more general harmonic families (e.g., Maass forms on higher-genus arithmetic quotients).
• Further refining the arithmetic analysis to obtain (either conditionally or unconditionally) higher proportions of critical line zeros, pushing towards the conjectured universality limits suggested by random matrix theory.

Conclusion

The paper achieves significant progress in the mean value and zero-distribution theory for families of automorphic $\mathrm{PGL}(2)$1-functions, providing strong unconditional results and a comprehensive framework for ALS in automorphic contexts without reliance on bounds toward the Ramanujan Conjecture. The techniques and results set a new standard for zero-density and mean value analysis in automorphic families, and the detailed arithmetic developed paves the way for future advances in analytic number theory.

Reference: "Critical Zeros and Unconditional Mean Value Theorems for twisted $\mathrm{PGL}(2)$2 and $\mathrm{PGL}(2)$3 $\mathrm{PGL}(2)$4-functions" (2607.00282).