Central values of additive twists of automorphic $L$-series

Abstract

We study series $L(x,s):=\sum_n ρ(n)e^{2πi n x}/n^s$, where $ρ(n)$ are Fourier coefficients of an automorphic form $ϕ$ for a discrete cofinite subgroup $Γ\subset \mathrm{SL}(2,\mathbb{R})$ and $x\in\mathbb{R}$. Let $\mathfrak{b}\in\mathbb{R}\cup\{\infty\}$ represent a cusp of $Γ$. Ordering the orbit $Γ\mathfrak{b}$ by increasing denominators, we prove that the values $L(x,1/2)$ for $x\inΓ\mathfrak{b}$, suitably normalized, become equidistributed according to a standard complex Gaussian law, under natural assumptions on the Laplace eigenvalue of $ϕ$. Previously this was known either when $Γ=\mathrm{SL}(2,\mathbb{Z})$ or when $ϕ$ is associated to a holomorphic cusp form. This applies in particular to central values at $s=1/2$ of additive twists of Maass form $L$-functions $\sum_{n\geq1}λ(n)e^{2πi n x}/n^s$, or to the Dirichlet-theta series $\sum_{n\geq1}χ(n)e^{2πi n^2 x}/n^s$.