Sparse Distribution of Coefficients of $\ell$-fold Product $L$-functions at Integers Represented by Quadratic Forms
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866917219336716288 |
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| author | Sharma, Anubhav Tripathi, Mohit Vaishya, Lalit |
| author_facet | Sharma, Anubhav Tripathi, Mohit Vaishya, Lalit |
| contents | Let $f \in S_{k}(Γ_{0}(N))$ be a normalized Hecke eigenform. We study the Fourier coefficients $λ_{f \otimes \cdots \otimes_{\ell} f}(n)$ of the $\ell$-fold product $L$-function for odd $\ell \ge 3$. Our focus is the distribution of this sequence over the sparse set of integers represented by a primitive, positive-definite binary quadratic form $Q$ of a fixed discriminant $D$. We establish an explicit upper bound for the summatory function of these coefficients, with dependencies on the weight, level, and discriminant. As a key application, we provide a bound for the first sign change of the sequence in this setting. We also generalize this result to find the first sign change among integers represented by any of the $h(D)$ forms of discriminant $D$, showing the bound improves as the class number increases. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_16352 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Sparse Distribution of Coefficients of $\ell$-fold Product $L$-functions at Integers Represented by Quadratic Forms Sharma, Anubhav Tripathi, Mohit Vaishya, Lalit Number Theory 11F30, 11F11, 11M06, 11N37 Let $f \in S_{k}(Γ_{0}(N))$ be a normalized Hecke eigenform. We study the Fourier coefficients $λ_{f \otimes \cdots \otimes_{\ell} f}(n)$ of the $\ell$-fold product $L$-function for odd $\ell \ge 3$. Our focus is the distribution of this sequence over the sparse set of integers represented by a primitive, positive-definite binary quadratic form $Q$ of a fixed discriminant $D$. We establish an explicit upper bound for the summatory function of these coefficients, with dependencies on the weight, level, and discriminant. As a key application, we provide a bound for the first sign change of the sequence in this setting. We also generalize this result to find the first sign change among integers represented by any of the $h(D)$ forms of discriminant $D$, showing the bound improves as the class number increases. |
| title | Sparse Distribution of Coefficients of $\ell$-fold Product $L$-functions at Integers Represented by Quadratic Forms |
| topic | Number Theory 11F30, 11F11, 11M06, 11N37 |
| url | https://arxiv.org/abs/2601.16352 |