Large sums of Hecke eigenvalues of holomorphic cusp forms

Abstract

Abstract Let f be a Hecke cusp form of weight k for the full modular group, and let { λ f ⁢ ( n ) } n ≥ 1 {\{\lambda_{f}(n)\}_{n\geq 1}} be the sequence of its normalized Fourier coefficients. Motivated by the problem of the first sign change of λ f ⁢ ( n ) {\lambda_{f}(n)} , we investigate the range of x (in terms of k) for which there are cancellations in the sum S f ⁢ ( x ) = ∑ n ≤ x λ f ⁢ ( n ) {S_{f}(x)=\sum_{n\leq x}\lambda_{f}(n)} . We first show that S f ⁢ ( x ) = o ⁢ ( x ⁢ log ⁡ x ) {S_{f}(x)=o(x\log x)} implies that λ f ⁢ ( n ) < 0 {\lambda_{f}(n)<0} for some n ≤ x {n\leq x} . We also prove that S f ⁢ ( x ) = o ⁢ ( x ⁢ log ⁡ x ) {S_{f}(x)=o(x\log x)} in the range log ⁡ x / log ⁡ log ⁡ k → ∞ {\log x/\log\log k\to\infty} assuming the Riemann hypothesis for L ⁢ ( s , f ) {L(s,f)} , and furthermore that this range is best possible unconditionally. More precisely, we establish the existence of many Hecke cusp forms f of large weight k, for which S f ⁢ ( x ) ≫ A x ⁢ log ⁡ x {S_{f}(x)\gg_{A}x\log x} , when x = ( log ⁡ k ) A {x=(\log k)^{A}} . Our results are GL 2 {\mathrm{GL}_{2}} analogues of work of Granville and Soundararajan for character sums, and could also be generalized to other families of automorphic forms.