The Second Moment of Sums of Hecke Eigenvalues I
Ned Carmichael
TL;DR
This paper analyzes the average size of short sums of Hecke eigenvalues $\lambda_f(n)$ over holomorphic cusp forms of large weight $k$, focusing on the second moment $\langle S(x,f)^2\rangle$ and the first moment $\langle S(x,f)\rangle$ as the interval length $x$ varies with respect to $k$. The authors combine the Petersson trace formula with Voronoï-type summation and Poisson summation to separate diagonal and off-diagonal contributions, and to study oscillatory integrals driven by $J_{k-1}$-type Bessel functions. They establish clear transitions in the average behavior: first and second moments exhibit square-root-like cancellation for $x$ up to about $k^2/(16\pi^2)$, with a noticeable shift near $x\approx k$ and a secondary transition in $x\approx k^2/(32\pi^2)$, $k^2/(16\pi^2)$ for the first moment, and a transition around $x\approx k/(4\pi)$ for the second moment, manifesting in a secondary term involving a piecewise logarithmic function $L$. The second part will extend these results to the regime $x>k^2$ where even smaller average sums emerge. This work advances understanding of fluctuations of Hecke eigenvalues in short intervals and lays groundwork for central-limit-type phenomena in these averages.
Abstract
Let $f$ be a Hecke cusp form of weight $k$ for $\mathrm{SL}_2(\mathbb{Z})$, and let $(λ_f(n))_{n\geq1}$ denote its (suitably normalised) 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 interesting 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.