# Nonvanishing of central L-values of Maass forms

Are there any results on the proportion of nonzero central L-values of Maass cusp forms? More precisely, I am looking for lower bounds for

$$\frac{\#\{\phi_j : \, L(1/2, \phi_j) \neq 0, \, \lambda_j \leq T\}}{\#\{\phi_j : \, \lambda_j \leq T\}}$$

as $T \rightarrow \infty$, where $\phi_j$ (with eigenvalue $\lambda_j$) is an orthonormal basis of Maass cusp forms for a congruence subgroup of $\mathrm{SL}_2(\mathbb{Z})$.

I ask this question as I found results of this kind for L-functions of holomorphic cusp forms in Iwaniec-Sarnak's work ''The non-vanishing of central values of automorphic L-functions and Landau-Siegel zeros'' and for Rankin-Selberg L-functions of Maass forms and a fixed holomorphic cusp form in Luo's work ''Nonvanishing of L-Values and the Weyl Law''.

## 2 Answers

For the full modular group $\mathrm{SL}_2(\mathbb{Z})$, Zhao Xu proved that a positive proportion of these $L$-values do not vanish, even when $\lambda_j$ is restricted to a short interval $\lambda_j\in[T-V,T+V]$ with $cT^{1/2}\log T\leq V\leq T$ (where $c>0$ is some large constant). See his paper: Nonvanishing of automorphic L-functions at special points, Acta Arith. 162 (2014), 309–335.

Added. This implies a positive proportion of nonvanishing for congruence subgroups $\Gamma_0(N)$ as well (assuming the nebentypus is trivial), because the spectrum of these include the spectrum of $\mathrm{SL}_2(\mathbb{Z})$ via oldforms. Probably Zhao Xu's proof can be extended to newforms of level $N$ (and any nebentypus) as well.

Apologies, I know this is a very late response to an old question. However, note that the paper referenced in the other answer shows that a positive proportion of the values $$L(\frac12 + it_j, u_j)$$ are nonvanishing in short intervals, where $$u_j$$ is a Maass cusp form with Laplace eigenvalue $$\frac14 + t_j^2$$, which has the "special point" $$\frac12 + it_j$$. I think the original question is about the central values $$L(\frac12, u_j)$$.

For $$SL(2, \mathbb{Z})$$, in Jeff Vanderkam's thesis he shows that at least $$\frac16$$ of $$L(\frac12, u_j)$$ are nonvanishing within a long interval; of course, $$L(\frac12, u_j) = 0$$ for odd $$u_j$$, so when restricting to even forms then the proportion is at least $$\frac13$$.

Shenhui Liu shows that a positive proportion of the central values are nonvanishing in short intervals–see his thesis or preprint https://arxiv.org/pdf/1702.07084.pdf.

The current best is due to Balkanova, Huang, and Södergren, who show that the proportion of nonvanishing at the central point (again, for even $$u_j$$) is at least $$\frac12$$–see their preprint https://arxiv.org/pdf/1810.07991.pdf.