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 $\asymp x$.
Categories: math.NT