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| Format: | Preprint |
| Published: |
2026
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| Online Access: | https://arxiv.org/abs/2601.17079 |
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| _version_ | 1866917324018155520 |
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| author | Venkatasubbareddy, K. |
| author_facet | Venkatasubbareddy, K. |
| contents | For an even integer $k\geq 2$, let $f$ be a primitive holomorphic cusp form of weight $k$ for the full modular group $SL(2,\mathbb{Z})$ and let $λ_{\rm{sym}^jf}(n)$ denote the $n^\text{th}$ normalized Fourier coefficient of the $j^{\text{th}}$ symmetric power $L$-function $L(s,{\rm{sym}}^j f)$. It has been an interesting problem to study the average behaviour of $λ_{\rm{sym}^jf}(n)$ and their higher powers, and many researchers in the literature have studied the sum \begin{equation*}
\sum_{n\leq x} λ_{\rm{sym}^j}^l(n), \end{equation*} for various values of $l$ and $j$. In this paper, we improve and generalize previously known results concerning the sum above for positive integers $l$ and $j$ such that $lj\geq 4$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_17079 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Generalization on the higher moments of the Fourier coefficients of symmetric power $L$-functions Venkatasubbareddy, K. Number Theory For an even integer $k\geq 2$, let $f$ be a primitive holomorphic cusp form of weight $k$ for the full modular group $SL(2,\mathbb{Z})$ and let $λ_{\rm{sym}^jf}(n)$ denote the $n^\text{th}$ normalized Fourier coefficient of the $j^{\text{th}}$ symmetric power $L$-function $L(s,{\rm{sym}}^j f)$. It has been an interesting problem to study the average behaviour of $λ_{\rm{sym}^jf}(n)$ and their higher powers, and many researchers in the literature have studied the sum \begin{equation*} \sum_{n\leq x} λ_{\rm{sym}^j}^l(n), \end{equation*} for various values of $l$ and $j$. In this paper, we improve and generalize previously known results concerning the sum above for positive integers $l$ and $j$ such that $lj\geq 4$. |
| title | Generalization on the higher moments of the Fourier coefficients of symmetric power $L$-functions |
| topic | Number Theory |
| url | https://arxiv.org/abs/2601.17079 |