The Second Moment of Sums of Hecke Eigenvalues II
Ned Carmichael
TL;DR
This paper analyzes the average behavior, over weight-$k$ Hecke eigenforms, of the short-interval sums $\mathcal{S}(x,f)=\sum_{x\le n\le 2x} \lambda_f(n)$. Using a sharp Voronoï summation formula, the Petersson trace formula, and meticulous oscillatory-integral analysis, the authors derive precise first and second moment formulas for $\langle \mathcal{S}(x,f)\rangle$ and $\langle \mathcal{S}(x,f)^2\rangle$ in the weight-aspect as $k\to\infty$, for a range of $x$ around and above the critical scale $k^2$. The main term of the second moment is $32\pi x^{1/2} \sum_{n\ge1} \frac{\Omega(n,x)^2}{n^{3/2}}$, with error terms that decay in $k$ and a subordinate diagonal/off-diagonal decomposition. The results reveal a transition in moment size: in the regime $x \ge k^2$, the second moment is of size roughly $x^{1/2}$, contrasting with the $x$-size in the slightly smaller regime studied in Part I, thereby sharpening our understanding of the fluctuation scale of $\mathcal{S}(x,f)$ in weight aspect. Overall, the paper provides a sharp, implementable framework for second-moment analysis of Hecke eigenvalue sums in the $k$-aspect, with explicit main-term structures governed by the oscillatory weights $\Omega(n,x)$.
Abstract
Let $f$ be a Hecke cusp form of weight $k$ for $\mathrm{SL}_2(\mathbb{Z})$, and let $(λ_f(n))_{n\geq 1}$ 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 $k$. It is proved that when the length of the sums $x$ is larger than $k^2$, the second moment is roughly of size $x^{1/2}$. This is in sharp contrast to the regime where $x$ is slightly smaller than $k^2$, where it was shown in preceding work (part I) that the second moment is of size $x$.