On the Effective Non-vanishing of Hecke--Maass L-functions at Special Points

2025/07/19 by Qi, Zhi
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2507.14566

Abstract

In this paper, we consider the non-vanishing problem for the family of special Hecke--Maass L-values L (1/2+it_f, f) with f (z) in an orthonormal basis of (even or odd) Hecke--Maass cusp forms of Laplace eigenvalue 1/4 + t_f² (t_f > 0). We prove that 33% of L (1/2+it_f, f) for t_f \leqslant T do not vanish as T → ∞. For comparison, it is known that the non-vanishing proportion is at least 25% for the central L-values L (1/2, f). Further, 33% may be raised to 50% conditionally on the generalized Riemann hypothesis. Moreover, we prove non-vanishing results on short intervals |t_f-T| \leqslant T^μ for any 0 < μ< 1. However, it is a curious case that the Riemann hypothesis does not yield better result for small 0 < μ\leqslant 1/2.

Citations

• On the Effective Non-vanishing of Rankin--Selberg L-functions at Special Points
• Moments of Central L-values for Maass Forms over Imaginary Quadratic Fields
• Low-lying zeros of L-functions for Maass forms over imaginary quadratic fields
• Non-vanishing of Maass form L-functions at the critical point
• Decoupling, exponential sums and the Riemann zeta function
• Low lying zeros of families of L-functions
• Zeros of principal L-functions and random matrix theory

Related

• Non-vanishing of Maass form L-functions at the critical point
• On the Effective Non-vanishing of Rankin--Selberg L-functions at Special Points
• Simultaneous non-vanishing of central values of GL(2) × GL(3) and GL(3) L-functions
• The Second Moment of GL₃ × GL₂ L-functions at Special Points
• Non-vanishing of L-functions associated to cusp forms of half-integral weight