First moment of Hecke eigenvalues at the integers represented by binary quadratic forms
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arXiv
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| Format: | Preprint |
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2024
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| author | Pandey, Manish Kumar vaishya, Lalit |
| author_facet | Pandey, Manish Kumar vaishya, Lalit |
| contents | In the article, we consider a question concerning the estimation of summatory function of the Fourier coefficients of Hecke eigenforms indexed by a sparse set of integers. In particular, we provide an estimate for the following sum;
\begin{equation*} \begin{split} S(f, \mathcal{Q}; X ) &:= \sideset{}{^{\flat }}\sum_{n= \mathcal{Q}(\underline{x}) \le X \atop \gcd(n,N) =1 } λ_{f}(n), \end{split}\end{equation*} where $\flat$ means that sum runs over the square-free positive integers, $λ_{f}(n)$ denotes the normalised $n^{\rm th}$ Fourier coefficients of a Hecke eigenform $f$ of integral weight $k$ for the congruence subgroup $Γ_{0}(N)$ and $\mathcal{Q}$ is a primitive integral positive-definite binary quadratic forms of fixed discriminant $D<0$ with the class number $h(D)=1$. As a consequence, we determine the size, in terms of conductor of associated $L$-function, for the first sign change of Hecke eigenvalues indexed by the integers which are represented by $\mathcal{Q}$. This work is an improvement and generalisation of the previous results. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2401_18055 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | First moment of Hecke eigenvalues at the integers represented by binary quadratic forms Pandey, Manish Kumar vaishya, Lalit Number Theory Primary 11F30, 11F11, 11M06, Secondary 11N37 In the article, we consider a question concerning the estimation of summatory function of the Fourier coefficients of Hecke eigenforms indexed by a sparse set of integers. In particular, we provide an estimate for the following sum; \begin{equation*} \begin{split} S(f, \mathcal{Q}; X ) &:= \sideset{}{^{\flat }}\sum_{n= \mathcal{Q}(\underline{x}) \le X \atop \gcd(n,N) =1 } λ_{f}(n), \end{split}\end{equation*} where $\flat$ means that sum runs over the square-free positive integers, $λ_{f}(n)$ denotes the normalised $n^{\rm th}$ Fourier coefficients of a Hecke eigenform $f$ of integral weight $k$ for the congruence subgroup $Γ_{0}(N)$ and $\mathcal{Q}$ is a primitive integral positive-definite binary quadratic forms of fixed discriminant $D<0$ with the class number $h(D)=1$. As a consequence, we determine the size, in terms of conductor of associated $L$-function, for the first sign change of Hecke eigenvalues indexed by the integers which are represented by $\mathcal{Q}$. This work is an improvement and generalisation of the previous results. |
| title | First moment of Hecke eigenvalues at the integers represented by binary quadratic forms |
| topic | Number Theory Primary 11F30, 11F11, 11M06, Secondary 11N37 |
| url | https://arxiv.org/abs/2401.18055 |