Mean square estimate for relatively short exponential sums involving Fourier coefficients of cusp forms

Abstract

when α ∈ R. This was an improvement over the classical result by Wilton [12]. By the Rankin–Selberg mean value theorem [10] this bound is the best possible in the general case, even though for some values of α it is possible to prove considerably better bounds. For instance, for rational values of α = h k , the classical bound is ≪ kM. However, the behavior of short sums, i.e., the sums over an interval [x, y], where y− x = o(x) is much less known. These sums have been investigated for instance by the author and Karppinen in [3]. Even though some of the bounds proved there are sharp, it is likely that in many cases, the actual bounds are much smaller than what have been proved. It is generally a very difficult question to prove good bounds for individual sums. It is much easier to consider the average behavior, namely, to bound expressions of the type