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| Format: | Preprint |
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2024
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| Accès en ligne: | https://arxiv.org/abs/2407.18259 |
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| _version_ | 1866916336875077632 |
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| author | Sankaranarayanan, Kampamolla Venkatasubbareddy Ayyadurai |
| author_facet | Sankaranarayanan, Kampamolla Venkatasubbareddy Ayyadurai |
| contents | Let $H_k$ be the set of all normalized primitive holomorphic cusp forms of even integral weight $k\geq 2$ for the full modular group $SL(2, \mathbb{Z})$, and let $j\geq 3$ be any fixed integer. For $f\in H_k$, we write $λ_{{\rm{sym}^j }f}(n)$ for the $n^\textit{th}$ normalized Fourier coefficient of $L(s,{\rm{sym}}^j f)$. In this article, we establish an asymptotic formula for the sum $$\begin{equation} \sum_{\substack{n=a_1^2+a_2^2+\ldots+a_6^2\leq x\\ \left(a_1,a_2,\ldots, a_6\right)\in \mathbb{Z}^6}} λ_{\rm{sym}^j f}^2(n), \end{equation}$$ with an improved error term. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_18259 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Higher symmetric power $L$-functions and their Fourier coefficients Sankaranarayanan, Kampamolla Venkatasubbareddy Ayyadurai Number Theory Let $H_k$ be the set of all normalized primitive holomorphic cusp forms of even integral weight $k\geq 2$ for the full modular group $SL(2, \mathbb{Z})$, and let $j\geq 3$ be any fixed integer. For $f\in H_k$, we write $λ_{{\rm{sym}^j }f}(n)$ for the $n^\textit{th}$ normalized Fourier coefficient of $L(s,{\rm{sym}}^j f)$. In this article, we establish an asymptotic formula for the sum $$\begin{equation} \sum_{\substack{n=a_1^2+a_2^2+\ldots+a_6^2\leq x\\ \left(a_1,a_2,\ldots, a_6\right)\in \mathbb{Z}^6}} λ_{\rm{sym}^j f}^2(n), \end{equation}$$ with an improved error term. |
| title | Higher symmetric power $L$-functions and their Fourier coefficients |
| topic | Number Theory |
| url | https://arxiv.org/abs/2407.18259 |