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

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

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 2


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.


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