The shifted convolution L-function for Maass forms

Let \(\Phi _1,\Phi _2\) ${\mathrm{\Phi }}_1,{\mathrm{\Phi }}_2$ be Maass forms for \(SL (2,{\mathbb {Z}})\) $SL(2,\mathbb{Z})$ with Fourier coefficients \(C_1(n),C_2(n)\) $C_1(n),C_2(n)$ . For a positive integer h the meromorphic continuation and growth in \(s\in {\mathbb {C}}\) $s\in \mathbb{C}$ (away from poles) of the shifted convolution L-function \(\begin{aligned} L_h(s,{\Phi _1,\Phi _2})\,:= \sum _{n \ne 0,-h} {C_1(n) C_2(n + h)} \cdot \big |n(n + h)\big |^{-\frac{1}{2}s} \end{aligned}\) $$\begin{array}{c}\hfill L_h(s,{\mathrm{\Phi }}_1,{\mathrm{\Phi }}_2):=\sum _{n\ne 0,-h}{C}_1(n)C_2(n+h)·|n(n+h){|}^{-\frac{1}{2}s}\end{array}$$ is obtained. For \(\textrm{Re}(s) > 0\) $\text{Re}(s)>0$ it is shown that the only poles are possible simple poles at \(\frac{1}{2} \pm ir_k\) $\frac{1}{2}\pm i{r}_k$ , where \(\tfrac{1}{4}+r_k^2\) $\frac{1}{4}+r_k^2$ are eigenvalues of the Laplacian. As an application we obtain, for \(T\rightarrow \infty \) $T\to \infty$ , the asymptotic formula \(\begin{aligned}&\underset{n \ne 0,-h}{\sum _{\sqrt{|n (n + h)|}<T} } \hspace{-5.0pt}{C_1(n) C_2(n + h)} \left( log \Big (\tfrac{T}{\sqrt{|n (n + h)|}}\,\Big )\right) ^{\frac{3}{2} + \varepsilon } \hspace{-7.0pt}\\ &\qquad =\; f_{{{\mathfrak {r}}_1,{\mathfrak {r}}_2,}h,\varepsilon }(T) \cdot T^{\frac{1}{2}} \; + \; {\mathcal {O}}\left( h^{1-\varepsilon } T^\varepsilon + h^{1 + \varepsilon } T^{-2 - 2\varepsilon } \right) , \end{aligned}\) $$\begin{array}{cc}& \underset{n\ne 0,-h}{\sum _{\sqrt{|n(n+h)|}<T}}{C}_1(n)C_2(n+h){\left(l,o,g,(,\frac{T}{\sqrt{|n(n+h)|}},,)\right)}^{\frac{3}{2}+\epsilon }\hfill \\ \hfill & =f_{{\mathfrak{r}}_1,{\mathfrak{r}}_2,h,\epsilon }(T)·T^{\frac{1}{2}}+\mathcal{O}\left(h^{1-\epsilon },T^{\epsilon },+,h^{1+\epsilon },T^{-2-2\epsilon }\right),\hfill \end{array}$$ where the function \(f_{{{\mathfrak {r}}_1,{\mathfrak {r}}_2,}h,\varepsilon }(T)\) $f_{{\mathfrak{r}}_1,{\mathfrak{r}}_2,h,\epsilon }(T)$ is given as an explicit spectral sum that satisfies the bound \(f_{{{\mathfrak {r}}_1,{\mathfrak {r}}_2,}h,\varepsilon }(T) \ll h^{\theta + \varepsilon }\) $f_{{\mathfrak{r}}_1,{\mathfrak{r}}_2,h,\epsilon }(T)\ll h^{\theta +\epsilon }$ . We also obtain a sharp bound for the above shifted convolution sum with sharp cutoff, i.e., without the smoothing weight \(\log (*)^{\frac{3}{2}+\varepsilon }\) $log(\ast )_{}^{\frac{3}{2}+\epsilon }$ with uniformity in the h aspect. Specifically, we show that for \(h < x^{\frac{1}{2} - \varepsilon }\) $h<x^{\frac{1}{2}-\epsilon }$ , \(\begin{aligned} {\sum _{\sqrt{|n (n + h)|} < x} C_1(n) C_2(n + h)} \ll h^{\frac{2}{3}\theta + \varepsilon }x^{\frac{2}{3} (1 + \theta ) + \varepsilon } + h^{\frac{1}{2} + \varepsilon }x^{\frac{1}{2} + 2\theta + \varepsilon }. \end{aligned}\) $$\begin{array}{c}\hfill \sum _{\sqrt{|n(n+h)|}<x}{C}_1(n)C_2(n+h)\ll h^{\frac{2}{3}\theta +\epsilon }{x}^{\frac{2}{3}(1+\theta )+\epsilon }+h^{\frac{1}{2}+\epsilon }{x}^{\frac{1}{2}+2\theta +\epsilon }.\end{array}$$