Mean square bound for Hecke–Maass L-functions in the range 2T≤t_φ≤T^{1+δ}

Prove that for any even Hecke–Maass cusp form φ on Γ=SL_2(Z) with spectral parameter t_φ and L-function L_φ(s)=∑_{n≥1}λ_φ(n)n^{-s}, there exists δ>0 such that whenever 2T≤t_φ≤T^{1+δ}, the mean square bound (1/T)∫_T^{2T}|L_φ(1/2+it)|^2 dt≪_ε t_φ^ε holds for every ε>0 as T→∞.

Background

This conjectural L2 estimate serves as a replacement for the Lindelöf hypothesis in the authors’ method to obtain sharp lower bounds on sign changes of cusp forms along geodesic segments. It is known for the shorter-range regime 2T>t_φ, but the regime 2T≤t_φ≤T{1+δ} remains out of reach.

The conjecture also appears in related contexts, such as restricted quantum unique ergodicity for Maass forms. Establishing it would remove a key conditional assumption in the paper’s results for cusp forms.

References

Let $\phi$ be a Maass form with spectral parameter $t_\phi$. There exists a $\delta>0$ such that, for $2T\le t_\phi \le T{1+\delta}$ and every $\vep>0$, we have \begin{align}\label{L cusp bound} \frac{1}{T}\int_T{2T} {L_{\phi}(\frac{1}{2}+it)}2 d t \ll t_\phi\vep, \end{align} as $T\to\infty$. Such an estimate clearly follows from the Lindel"of hypothesis, and we note that for the range $2T > t_\phi$ the estimate L cusp bound is known (see Section 6.1). While it is possible that our range $2T\le t_\phi \le T{1+\delta}$ is also within reach of current technology we were not able to establish it and thus leave it as an open conjecture.

— Sign changes along geodesics of modular forms (2409.17248 - Kelmer et al., 25 Sep 2024) in Conjecture \ref{con:L2}, Subsection “Preparation for Maass forms”