On shifted convolution sums of $\mathrm{GL}(3)$-Fourier coefficients with an average over shifts
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| Format: | Preprint |
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2025
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| _version_ | 1866911216983605248 |
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| author | Pal, Ritwik Pal, Sampurna |
| author_facet | Pal, Ritwik Pal, Sampurna |
| contents | Let $F$ be a Hecke-Maass cusp form for $\mathrm{SL}_3(\mathbb{Z})$ and $A(m,n)$ be its normalized Fourier coefficients. Let $V$ be a smooth function, compactly supported on $[1,2]$ and satisfying $V(y)^{j} \ll_j y^{-j}$ for any $j \in \mathbb{N} \cup \{0\}$. In this article we prove a power-saving upper bound for the `average' shifted convolution sum \begin{equation*} \sum_{h}\sum_{n}A(1,n)A(1,n+h)V\left(\frac{n}{N}\right)V\left(\frac{h}{H}\right), \end{equation*} for the range $N^{1/2-\varepsilon} \geq H \geq N^{1/6+ \varepsilon}$, for any $\varepsilon >0$. This is an improvement over the previously known range $N^{1/2-\varepsilon} \geq H \geq N^{1/4+ \varepsilon}$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_15799 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On shifted convolution sums of $\mathrm{GL}(3)$-Fourier coefficients with an average over shifts Pal, Ritwik Pal, Sampurna Number Theory 11F66 (primary), 11M41 (secondary) Let $F$ be a Hecke-Maass cusp form for $\mathrm{SL}_3(\mathbb{Z})$ and $A(m,n)$ be its normalized Fourier coefficients. Let $V$ be a smooth function, compactly supported on $[1,2]$ and satisfying $V(y)^{j} \ll_j y^{-j}$ for any $j \in \mathbb{N} \cup \{0\}$. In this article we prove a power-saving upper bound for the `average' shifted convolution sum \begin{equation*} \sum_{h}\sum_{n}A(1,n)A(1,n+h)V\left(\frac{n}{N}\right)V\left(\frac{h}{H}\right), \end{equation*} for the range $N^{1/2-\varepsilon} \geq H \geq N^{1/6+ \varepsilon}$, for any $\varepsilon >0$. This is an improvement over the previously known range $N^{1/2-\varepsilon} \geq H \geq N^{1/4+ \varepsilon}$. |
| title | On shifted convolution sums of $\mathrm{GL}(3)$-Fourier coefficients with an average over shifts |
| topic | Number Theory 11F66 (primary), 11M41 (secondary) |
| url | https://arxiv.org/abs/2510.15799 |