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arXiv · 2502.03436

The Second Moment of Sums of Hecke Eigenvalues II

Ned Carmichael
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Abstract

Let $f$ be a holomorphic Hecke cusp form of weight $k$ for $\mathrm{SL}_2(\mathbb{Z})$, and let $(\lambda_f(n))_{n\geq 1}$ denote its sequence of normalised Hecke eigenvalues. We compute the first and second moments of the sums $S(x,f)=\sum_{x\leq n\leq 2x} \lambda_f(n)$, on average over forms $f$ of large weight $k$. In the range $k^2/(8\pi^2)\leq x\leq k^{12/5-\epsilon}$, 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 $\asymp x$.

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Ned Carmichael. 2025-02-05. The Second Moment of Sums of Hecke Eigenvalues II. https://arxiv.org/abs/2502.03436

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