On the mean square of short exponential sums related to cusp forms

Abstract

The purpose of the article is to estimate the mean square of a squareroot length exponential sum of Fourier coefficients of a holomorphic cusp form. 1≤n≤M a(n)e(nα), where α is a real number, have been widely studied. See e.g. Wilton [11] and Jutila [9]. Short sums M ≤n≤M +∆ a(n)e(nα), where ∆ ≪ M 3/4 have been studied for instance in [3] and [4]. However, it seems that very short sums, in particular, sums with ∆ ≍ M 1/2 seem to be extremely difficult to treat, even though this is an important special case. According to the results in [1] and the computer data in [2], it is plausible to believe the correct upper bound to be M ≤n≤M + √ M a(n)e(nα) ≪ M 1/4+ε .

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

“…The main advantage in this theorem is the relatively short averaging interval compared to the length of the sum, when the value of k is small. The averaging interval is actually similar to the one in [2]. However, in the current paper, the length of the sum can be much longer than the averaging interval unlike there.…”

“…Questions closely connected to the topic of the current paper have also been dealt in [5], where Ivić proves the asymptotic result for α = 0, y ≪ √ x and ∆ = M, and in [8], where Jutila proved an asymptotic result for a mean-square involving a sum of values of the divisor function with y ≪ x 1/2 and ∆ ≫ M 1/2 . In the case y = √ x an exponential sum involving Fourier coefficients of a cusp form was dealt in [2]. There the averaging interval depended on the exponential twist similarly as in the current paper.…”

On Short Sums Involving Fourier Coefficients of Maass Forms

“…The average behaviour of short rationally additively twisted exponential sums weighted by Fourier coefficients of holomorphic cusp forms has been studied e.g. by Jutila [15], Ernvall-Hytönen [2,3], and Vesalainen [28]. In the higher rank setting, the mean square of long rationally additively twisted sums involving Fourier coefficients of GL(3) Maass cusp forms has been considered in [13].…”

On sums involving Fourier coefficients of Maass forms for $\mathrm{SL}(3,\mathbb Z)$

“…for any prime p, where P n is the polynomial given by (2). Hence, by using the explicit description of P n (α Therefore m≤x |A(m, 1, ..., 1)…”

“…Note that the condition y < (nY /2) n is satisfied by the choice of Y . The contribution coming from the error terms of Corollary 5 can be estimated as follows by using partial summation together with (7) and recalling the fact that Y ≍ X (1+θ)/n : Let us fix a smooth compactly supported non-negative weight function w majorising the characteristic function of the interval [1,2]. Now we simply compute:…”