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| Glavni autor: | |
|---|---|
| Format: | Preprint |
| Izdano: |
2025
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| Teme: | |
| Online pristup: | https://arxiv.org/abs/2502.03436 |
| Oznake: |
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- 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 $\asymp x$.