Year: 2025
Type: preprint
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 $1/4 + t_f^2$ ($t_f > 0$). We prove that 20% of $L (1/2+it_f, f)$ for $ t_f \leqslant T$ do not vanish as $T \rightarrow \infty$. For comparison, it is known that the non-vanishing proportion is at least 25% for the central $L$-values $L (1/2, f)$. Further, 20% may be raised to 50% conditionally on the generalized Riemann hypothesis. Moreover, we prove non-vanishing results on the short interval $|t_f-T| \leqslant T^μ$ for any $0 < μ< 1$.
Source: arXiv (Cornell University)
Author: Qi, Zhi
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W4417429532
Cites:
Cited by:
Related to: