Short-interval averages of sums of Fourier coefficients of cusp forms

Abstract

Let f be a weight k holomorphic cusp form of level one, and let Sf(n) denote the sum of the first n Fourier coefficients of f. In analogy with Dirichlet's divisor problem, it is conjectured that Sf(X)≪Xk−12+14+ϵ. Understanding and bounding Sf(X) has been a very active area of research. The current best bound for individual Sf(X) is Sf(X)≪Xk−12+13(log⁡X)−0.1185 from Wu [17].Chandrasekharan and Narasimhan [2] showed that the Classical Conjecture for Sf(X) holds on average over intervals of length X. Jutila [10] improved this result to show that the Classical Conjecture for Sf(X) holds on average over short intervals of length X34+ϵ. Building on the results and analytic information about ∑|Sf(n)|2n−(s+k−1) from our recent work [9], we further improve these results to show that the Classical Conjecture for Sf(X) holds on average over short intervals of length X23(log⁡X)16.