The Second Moment of Sums of Hecke Eigenvalues II

Abstract

Let $f$ be a holomorphic Hecke cusp form of weight $k$ for ${\mathrm{SL}}_2(\mathbb{Z})$, and let $({\lambda }_f(n){)}_{n\ge 1}$ denote its sequence of normalised Hecke eigenvalues. We compute the first and second moments of the sums $S(x,f)={\sum }_{x\le n\le 2x}{\lambda }_f(n)$, on average over forms $f$ of large weight $k$. In the range $k^2/(8{\pi }^2)\le x\le k^{12/5-\epsilon }$, 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 $x\le k^{2-o(1)}$, where the second moment was shown in preceding work (part I) to be of size $\asymp x$.

32 pages. Minor corrections and improved structure