Exponential Sums Twisted By Fourier Coefficients Of Automorphic Cusp Forms For SL(2,Z)

Let{an} be an arithmetic sequence of order1. Consider the exponential sum twisted by an: This problem was first considered by Hardy and Littlewood in1914. The triv-ial bound for S(α, X) uniformly in a is O(X1+ε), while the best possible bound is O(X1/2+ε).When an are the normalized Fourier coefficients of a modular or automor-phic cusp form/for SL(2,Z). many authors studied the above exponential sum. Let f be an automorphic cusp form for SL(2.Z) of weight k. Its nor-malized Fourier coefficients λf(n) are. defined by In1929, Wilton proved that holds uniformly in α, this is the best possible uniform bound. However, this bound can be improved dramatically for some special a.Estimates for sums above have been connected to many important prob-lems in number theory. For instance, using Wilton’s bound. Titchmarsh achieved the correct bound for the second moment of L-function L(s)=In this paper, we consider estimates for a smoothly weighted exponential sum For certain numbers α, we will prove that the smooth sum admitted a bound better than O(X1/2+ε). The main techniques used in this paper include Dirich-let’s rational approximation of real numbers, a Voronoi summation formula and the asymptotic expansion for Bessel functions. Theorem1.1Fix φ(x)∈C∞(0.+∞) which is supported on [a.b]. Let a=a/q+λ, with (a,q)=1such that|λ|<1/X, q2<X1-ε. Then we have: for any II>0.When a is a rational number, we can get the following result by Theorem1.1immediately. Corrollary1.1Fix φ(x)∈C∞(0,+∞) which is supported on [a, b]. Let a/q be a rational number with (a,q)=1. If q2<X1-ε, then: for any H>0. In particular, when q=1, we obtain for any II>0.For an irrational number α, if there exists a smallest number Ï„(α), such that for any μ>Ï„(α),|α-a/q|<q-u has only finitely many solutions, then Ï„(α) is called the approximation exponent of a. When Ï„(α)>2, we can derive the following result from Theorem1.1. Corrollary1.2Let α be a transcendental number. If Ï„(α)>2, there exists a sequence Xkâ†'∞such that for anv H>0.