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| Format: | Preprint |
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2023
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| Online Access: | https://arxiv.org/abs/2311.06587 |
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| _version_ | 1866909291877761024 |
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| author | Goldfeld, Dorian Hinkle, Gerhardt Hoffstein, Jeffrey |
| author_facet | Goldfeld, Dorian Hinkle, Gerhardt Hoffstein, Jeffrey |
| contents | 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^{-2 - 2\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 $h < x^{\frac{1}{2} - \varepsilon}$,
\[
{\sum_{\sqrt{|n (n + h)|} < x} C_1(n) C_2(n + h)} \ll h^{\frac{2}{3}θ+ \varepsilon}x^{\frac{2}{3} (1 + θ) + \varepsilon} + h^{\frac{1}{2} + \varepsilon}x^{\frac{1}{2} + 2θ+ \varepsilon}.
\] |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2311_06587 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | The shifted convolution L-function for Maass forms Goldfeld, Dorian Hinkle, Gerhardt Hoffstein, Jeffrey Number Theory 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^{-2 - 2\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 $h < x^{\frac{1}{2} - \varepsilon}$, \[ {\sum_{\sqrt{|n (n + h)|} < x} C_1(n) C_2(n + h)} \ll h^{\frac{2}{3}θ+ \varepsilon}x^{\frac{2}{3} (1 + θ) + \varepsilon} + h^{\frac{1}{2} + \varepsilon}x^{\frac{1}{2} + 2θ+ \varepsilon}. \] |
| title | The shifted convolution L-function for Maass forms |
| topic | Number Theory |
| url | https://arxiv.org/abs/2311.06587 |