The shifted convolution L-function for Maass forms
Let \(\Phi_1,\Phi_2\) 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,{\Phi_1,\Phi_2})\, := \sum_{n \ne...
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| description | Let \(\Phi_1,\Phi_2\) 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 $$L_h(s,{\Phi_1,\Phi_2})\, := \sum_{n \neq 0,-h} {C_1(n) C_2(n + h)} \cdot \big|n(n + h)\big|^{-\frac{1}{2}s}$$ 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)|} |
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| fullrecord | <record><control><sourceid>proquest</sourceid><recordid>TN_cdi_proquest_journals_2889792041</recordid><sourceformat>XML</sourceformat><sourcesystem>PC</sourcesystem><sourcerecordid>2889792041</sourcerecordid><originalsourceid>FETCH-proquest_journals_28897920413</originalsourceid><addsrcrecordid>eNpjYuA0MjY21LUwMTLiYOAtLs4yMDAwMjM3MjU15mQwCslIVSjOyEwrSU1RSM7PK8vPKS3JzM9T8NFNK81LBjPT8osUfBMTi4tBrNxiHgbWtMSc4lReKM3NoOzmGuLsoVtQlF9YmlpcEp-VX1qUB5SKN7KwsDS3NDIwMTQmThUA4Acz5Q</addsrcrecordid><sourcetype>Aggregation Database</sourcetype><iscdi>true</iscdi><recordtype>article</recordtype><pqid>2889792041</pqid></control><display><type>article</type><title>The shifted convolution L-function for Maass forms</title><source>Free E- Journals</source><creator>Goldfeld, Dorian ; Hinkle, Gerhardt ; Hoffstein, Jeffrey</creator><creatorcontrib>Goldfeld, Dorian ; Hinkle, Gerhardt ; Hoffstein, Jeffrey</creatorcontrib><description>Let \(\Phi_1,\Phi_2\) 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 $$L_h(s,{\Phi_1,\Phi_2})\, := \sum_{n \neq 0,-h} {C_1(n) C_2(n + h)} \cdot \big|n(n + h)\big|^{-\frac{1}{2}s}$$ 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 \(f_{{\mathfrak r_1,\mathfrak r_2,}h,\varepsilon}(T) \ll h^{\theta + \varepsilon}\). 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 \(h < x^{\frac{1}{2} - \varepsilon}\), \[ {\sum_{\sqrt{|n (n + h)|} < x} C_1(n) C_2(n + h)} \ll h^{\frac{2}{3}\theta + \varepsilon}x^{\frac{2}{3} (1 + \theta) + \varepsilon} + h^{\frac{1}{2} + \varepsilon}x^{\frac{1}{2} + 2\theta + \varepsilon}. \]</description><identifier>EISSN: 2331-8422</identifier><language>eng</language><publisher>Ithaca: Cornell University Library, arXiv.org</publisher><subject>Convolution ; Eigenvalues ; Poles</subject><ispartof>arXiv.org, 2024-08</ispartof><rights>2024. This work is published under http://arxiv.org/licenses/nonexclusive-distrib/1.0/ (the “License”). Notwithstanding the ProQuest Terms and Conditions, you may use this content in accordance with the terms of the License.</rights><oa>free_for_read</oa><woscitedreferencessubscribed>false</woscitedreferencessubscribed></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><link.rule.ids>776,780</link.rule.ids></links><search><creatorcontrib>Goldfeld, Dorian</creatorcontrib><creatorcontrib>Hinkle, Gerhardt</creatorcontrib><creatorcontrib>Hoffstein, Jeffrey</creatorcontrib><title>The shifted convolution L-function for Maass forms</title><title>arXiv.org</title><description>Let \(\Phi_1,\Phi_2\) 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 $$L_h(s,{\Phi_1,\Phi_2})\, := \sum_{n \neq 0,-h} {C_1(n) C_2(n + h)} \cdot \big|n(n + h)\big|^{-\frac{1}{2}s}$$ 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 \(f_{{\mathfrak r_1,\mathfrak r_2,}h,\varepsilon}(T) \ll h^{\theta + \varepsilon}\). 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 \(h < x^{\frac{1}{2} - \varepsilon}\), \[ {\sum_{\sqrt{|n (n + h)|} < x} C_1(n) C_2(n + h)} \ll h^{\frac{2}{3}\theta + \varepsilon}x^{\frac{2}{3} (1 + \theta) + \varepsilon} + h^{\frac{1}{2} + \varepsilon}x^{\frac{1}{2} + 2\theta + \varepsilon}. \]</description><subject>Convolution</subject><subject>Eigenvalues</subject><subject>Poles</subject><issn>2331-8422</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2024</creationdate><recordtype>article</recordtype><sourceid>ABUWG</sourceid><sourceid>AFKRA</sourceid><sourceid>AZQEC</sourceid><sourceid>BENPR</sourceid><sourceid>CCPQU</sourceid><sourceid>DWQXO</sourceid><recordid>eNpjYuA0MjY21LUwMTLiYOAtLs4yMDAwMjM3MjU15mQwCslIVSjOyEwrSU1RSM7PK8vPKS3JzM9T8NFNK81LBjPT8osUfBMTi4tBrNxiHgbWtMSc4lReKM3NoOzmGuLsoVtQlF9YmlpcEp-VX1qUB5SKN7KwsDS3NDIwMTQmThUA4Acz5Q</recordid><startdate>20240820</startdate><enddate>20240820</enddate><creator>Goldfeld, Dorian</creator><creator>Hinkle, Gerhardt</creator><creator>Hoffstein, Jeffrey</creator><general>Cornell University Library, arXiv.org</general><scope>8FE</scope><scope>8FG</scope><scope>ABJCF</scope><scope>ABUWG</scope><scope>AFKRA</scope><scope>AZQEC</scope><scope>BENPR</scope><scope>BGLVJ</scope><scope>CCPQU</scope><scope>DWQXO</scope><scope>HCIFZ</scope><scope>L6V</scope><scope>M7S</scope><scope>PIMPY</scope><scope>PQEST</scope><scope>PQQKQ</scope><scope>PQUKI</scope><scope>PRINS</scope><scope>PTHSS</scope></search><sort><creationdate>20240820</creationdate><title>The shifted convolution L-function for Maass forms</title><author>Goldfeld, Dorian ; Hinkle, Gerhardt ; Hoffstein, Jeffrey</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-proquest_journals_28897920413</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2024</creationdate><topic>Convolution</topic><topic>Eigenvalues</topic><topic>Poles</topic><toplevel>online_resources</toplevel><creatorcontrib>Goldfeld, Dorian</creatorcontrib><creatorcontrib>Hinkle, Gerhardt</creatorcontrib><creatorcontrib>Hoffstein, Jeffrey</creatorcontrib><collection>ProQuest SciTech Collection</collection><collection>ProQuest Technology Collection</collection><collection>Materials Science & Engineering Collection</collection><collection>ProQuest Central (Alumni Edition)</collection><collection>ProQuest Central UK/Ireland</collection><collection>ProQuest Central Essentials</collection><collection>ProQuest Central</collection><collection>Technology Collection</collection><collection>ProQuest One Community College</collection><collection>ProQuest Central Korea</collection><collection>SciTech Premium Collection</collection><collection>ProQuest Engineering Collection</collection><collection>Engineering Database</collection><collection>Publicly Available Content Database</collection><collection>ProQuest One Academic Eastern Edition (DO NOT USE)</collection><collection>ProQuest One Academic</collection><collection>ProQuest One Academic UKI Edition</collection><collection>ProQuest Central China</collection><collection>Engineering Collection</collection></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Goldfeld, Dorian</au><au>Hinkle, Gerhardt</au><au>Hoffstein, Jeffrey</au><format>book</format><genre>document</genre><ristype>GEN</ristype><atitle>The shifted convolution L-function for Maass forms</atitle><jtitle>arXiv.org</jtitle><date>2024-08-20</date><risdate>2024</risdate><eissn>2331-8422</eissn><abstract>Let \(\Phi_1,\Phi_2\) 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 $$L_h(s,{\Phi_1,\Phi_2})\, := \sum_{n \neq 0,-h} {C_1(n) C_2(n + h)} \cdot \big|n(n + h)\big|^{-\frac{1}{2}s}$$ 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 \(f_{{\mathfrak r_1,\mathfrak r_2,}h,\varepsilon}(T) \ll h^{\theta + \varepsilon}\). 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 \(h < x^{\frac{1}{2} - \varepsilon}\), \[ {\sum_{\sqrt{|n (n + h)|} < x} C_1(n) C_2(n + h)} \ll h^{\frac{2}{3}\theta + \varepsilon}x^{\frac{2}{3} (1 + \theta) + \varepsilon} + h^{\frac{1}{2} + \varepsilon}x^{\frac{1}{2} + 2\theta + \varepsilon}. \]</abstract><cop>Ithaca</cop><pub>Cornell University Library, arXiv.org</pub><oa>free_for_read</oa></addata></record> |
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| subjects | Convolution Eigenvalues Poles |
| title | The shifted convolution L-function for Maass forms |
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