The Second Moment of Sums of Hecke Eigenvalues II
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866910041844482048 |
|---|---|
| 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 $(λ_f(n))_{n\geq 1}$ denote its sequence of normalised 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 range $k^2/(8π^2)\leq x\leq k^{12/5-ε}$, the size of the second moment lies between $x^{1/2-o(1)}$ and $x^{1/2}$. This is in sharp contrast to the regime $x\leq k^{2-o(1)}$, where the second moment was shown in preceding work (part I) to be of size $\asymp x$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_03436 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Second Moment of Sums of Hecke Eigenvalues II 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 $(λ_f(n))_{n\geq 1}$ denote its sequence of normalised 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 range $k^2/(8π^2)\leq x\leq k^{12/5-ε}$, the size of the second moment lies between $x^{1/2-o(1)}$ and $x^{1/2}$. This is in sharp contrast to the regime $x\leq k^{2-o(1)}$, where the second moment was shown in preceding work (part I) to be of size $\asymp x$. |
| title | The Second Moment of Sums of Hecke Eigenvalues II |
| topic | Number Theory 11F30 (Primary) 11N37, 11F11 (Secondary) |
| url | https://arxiv.org/abs/2502.03436 |