The shifted convolution L-function for Maass forms

Abstract

Let $Φ_1,Φ_2$ be Maass forms for $\text{SL}(2,\mathbb Z)$ with Fourier coefficients $C_1(n),C_2(n)$. For a positive integer $h$ the meromorphic continuation and growth in $s\in\mathbb C$ (away from poles) of the shifted convolution L-function $$L_h(s,{Φ_1,Φ_2})\, := \sum_{n \neq 0,-h} {C_1(n) C_2(n + h)} \cdot \big|n(n + h)\big|^{-\frac{1}{2}s}$$ is obtained. For ${\rm Re}(s) > 0$ it is shown that the only poles are possible simple poles at $\frac{1}{2} \pm ir_k$, where $\tfrac14+r_k^2$ are eigenvalues of the Laplacian. As an application we obtain, for $T\to\infty$, the asymptotic formula \begin{align*} & \underset{n \neq 0,-h}{\sum_{\sqrt{|n (n + h)|}<T} } \hskip-5pt{C_1(n) C_2(n + h)} \left(\text{log}\Big(\tfrac{T}{\sqrt{|n (n + h)|}}\,\Big)\right)^{\frac{3}{2} + \varepsilon} \hskip-7pt =\; f_{{\mathfrak r_1,\mathfrak r_2,}h,\varepsilon}(T) \cdot T^{\frac{1}{2}} \; + \; \mathcal O\left( h^{1-\varepsilon} T^\varepsilon + h^{1 + \varepsilon} T^{-1 - \varepsilon} \right), \end{align*} where the function $f_{{\mathfrak r_1,\mathfrak r_2,}h,\varepsilon}(T)$ is given as an explicit spectral sum that satisfies the bound $f_{{\mathfrak r_1,\mathfrak r_2,}h,\varepsilon}(T) \ll h^{θ+ \varepsilon}$. We also obtain a sharp bound for the above shifted convolution sum with sharp cutoff, i.e., without the smoothing weight $\log(*)^{\frac32+\varepsilon}$ with uniformity in the $h$ aspect. Specifically, we show that for $h < x^{\frac{1}{2} - \varepsilon}$, \[ {\sum_{\sqrt{|n (n + h)|} < x} C_1(n) C_2(n + h)} \ll h^{\frac{2}{3}θ+ \varepsilon}x^{\frac{2}{3} (1 + θ) + \varepsilon} + h^{\frac{1}{2} + \varepsilon}x^{\frac{1}{2} + 2θ+ \varepsilon}. \]