On the Effective Non-vanishing of Rankin--Selberg L-functions at Special Points

2025/06/10 by Qi, Zhi · 1 citation
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2506.08546

Abstract

Let Q (z) be a holomorphic Hecke cusp newform of square-free level and u_j (z) traverse an orthonormal basis of Hecke--Maass cusp forms of full level. Let 1/4 + t_j² be the Laplace eigenvalue u_j (z). In this paper, we prove that there is a constant γ(Q) expressed as a certain Euler product associated to Q such that at least γ(Q) / 11 of the Rankin--Selberg special L-values L (1/2+it_j, Q ⊗ u_j) for t_j \leqslant T do not vanish as T → ∞. Further, we show that the non-vanishing proportion is at least γ(Q) ⋅ (4μ-3) / (4μ+7) on the short interval |t_j - T| \leqslant T^μ for any 3/4 < μ< 1.

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

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