The Second Moment of Sums of Hecke Eigenvalues I
Abstract
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 Hecke eigenvalues. We compute the first and second moments of the sums $\mathcal {S}(x,f)=\sum _{x< n\le 2x}\lambda _f(n)$, 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 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 $\mathcal {S}(x,f)$ becomes dramatically smaller.
Fields of Study & Subjects
Analytic Number Theory Research
Publication details
- Journal / source
- The Quarterly Journal of Mathematics
- Volume
- 77
- Issue
- 2
- Pages
- 423–454
- Publication date
- 16 March 2026
- Language
- EN