The Second Moment of Sums of Hecke Eigenvalues I
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866910046677368832 |
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| author | Carmichael, Ned |
| author_facet | Carmichael, Ned |
| contents | Let $f$ be a holomorphic Hecke cusp form of weight $k$ for $\mathrm{SL}_2(\mathbb{Z})$, and let $(λ_f(n))_{n\geq1}$ denote its sequence of Hecke eigenvalues. We compute the first and second moments of the sums $S(x,f)=\sum_{x\leq n\leq 2x}λ_f(n)$, on average over forms $f$ of large weight $k$, in the regime where the length of the sums $x$ is smaller than $k^2$. We observe transitions in the size of the sums when $x\approx k$ and $x\approx k^2$. In subsequent work (part II), it will be shown that once $x$ is larger than $k^2$ (where the latter transition occurs), the average size of the sums $S(x,f)$ becomes dramatically smaller. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_03437 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Second Moment of Sums of Hecke Eigenvalues I Carmichael, Ned Number Theory 11F30 (Primary) 11N37, 11F11 (Secondary) Let $f$ be a holomorphic Hecke cusp form of weight $k$ for $\mathrm{SL}_2(\mathbb{Z})$, and let $(λ_f(n))_{n\geq1}$ denote its sequence of Hecke eigenvalues. We compute the first and second moments of the sums $S(x,f)=\sum_{x\leq n\leq 2x}λ_f(n)$, on average over forms $f$ of large weight $k$, in the regime where the length of the sums $x$ is smaller than $k^2$. We observe transitions in the size of the sums when $x\approx k$ and $x\approx k^2$. In subsequent work (part II), it will be shown that once $x$ is larger than $k^2$ (where the latter transition occurs), the average size of the sums $S(x,f)$ becomes dramatically smaller. |
| title | The Second Moment of Sums of Hecke Eigenvalues I |
| topic | Number Theory 11F30 (Primary) 11N37, 11F11 (Secondary) |
| url | https://arxiv.org/abs/2502.03437 |