Landau-Siegel zeros of Rankin-Selberg $L$-functions

TL;DR

This work establishes robust zero-free regions for Rankin–Selberg and triple product L-functions, extending Landau–Siegel zero elimination to new families where modularity is not yet known. The authors construct carefully weighted Dirichlet products D(s) with nonnegative Dirichlet coefficients, leveraging a GHL-type framework to convert potential real zeros into contradictions with pole structures, even when modularity is incomplete. Central to the approach is a detailed case analysis guided by Polyhedral type (dihedral, tetrahedral, octahedral) classifications, symmetric power lifts, and isobaric decompositions, which reduce complex high-rank L-functions to products of lower-rank factors with known zero-free behavior. The results yield explicit no-exceptional-zero statements for a wide range of L-functions, including symmetric-square and higher symmetric powers, mixed Rankin–Selberg products, and certain triple products, thereby strengthening unconditional prime-distribution results and broadening zero-free regions beyond previously modular cases. The findings have broad implications for analytic number theory, improving unconditional bounds in prime counting and enabling further progress in the Langlands program where modularity remains unproven in full generality.

Abstract

We establish standard zero-free regions with no exceptional Landau-Siegel zeros for Rankin-Selberg $L$-functions and triple product $L$-functions in several new families for which modularity is not yet known.

Theorems & Definitions (56)

• Proposition 1.1
• Proposition 1.2: Thorner_Siegel
• Theorem 1.3
• Remark 1.4
• Theorem 1.5
• Remark 1.6
• Theorem 1.7
• Theorem 2.1
• Lemma 2.2
• Lemma 2.3