On Exponential Sums Twisted By Fourier Coefficients Of Maass Cusp Forms For SL(n, Z) And Generalizations

In analytic number theory, the Fourier coefficients of cusp forms are fas-cinating and important arithmetic functions that have been studied extensive-ly in various aspects. One open problem is Ramanujan-Petersson conjecture which predicts that the n-th normalized Fourier coefficient of a cusp form is bounded by nε, for any ε>0sufficiently small. In the case of holomorphic cusp forms of SL(2, Z), this was proved by Deligne [5] in1974as a conse-quence of his proof of Weil conjecture. However, other cases are still open. But the mean values imply Ramanujan-Petersson hoids. For example, let f be a Maass cusp form for SL(2, Z), then where λf(n) is the normalized n-th Fourier coefficient of f([10]). It is apparent from this that the Fourier coefficient of cusp forms has strong oscil-latory properties. So although we can’t proof the upper bound of Ramanujan-Petersson conjecture for individual Fourier coefficient, the mean values are good and have some applications.Recently, Ren and Ye ([26]) consider the sum where λf(n) is the normalized n-th Fourier coefficient of a holomorphic cusp form f weighted k for SL(2,Z). They proved when β≠1/2, the sum is bounded by Oα(Xmax(β,1/2-β/4)+ε). And when β=1/2and|α|is close to2(?)q for q∈Z+, there is an asymptotic formula. For Maass cusp forms for SL(2, Z),[26] did not give out the estimat. In this thesis, we study the estimates for the exponential sum where as usual, e(z)=e2πiz, and λg(n) are the normalized Fourier coeffi-cients of a Maass cusp form for SL(2, Z).Our results are as the following.Theorem1.1Let X>1,0<β <1.If|α|βXβ<(?)X/2, then If|α|βXβ≥(?)X/2, then where εα,q=1or0according to if there exists a positive integer q satisfying (|α|-2(?)q|≤X-1/2with1≤|α|<(?)X, and In particular, for any integer1≤q<X/4, we haveWe note that, the upper bound in Theorem1.1is X71/192+ε, which is depends on the Fourier coefficient of Hecke-Maass cusp forms. For holomorphic forms, the Ramanujan-Petersson conjecture λf(n)<<nε for any ε>0is true, while for Maass cusp cusp forms we can only use the bound λg(n)<<nθ where the best result so far is θ=7/64+ε for arbitrary ε>0(see [18]). But if Ramanujan-Petersson conjecture of Maass cusp forms holds, then we can get upper bound as better as that of holomorphic cusp forms. To prove the theorem, we will use a strategy used by Ren and Ye [26]. The input includes Voronoi’s summation formulas and asymptotic expansions of Bessel functions.It is apparent from the method for proof for Theorem1.1that some number-theoretic functions which Voronoi-type formula can be established can be consider in exponential sums, and looks like has the bound and asymptotic formula like the above. In this thesis, we consider the divisor function τ(n) and the representation function τ(n) in the circle problem which is defined as and have the following results.Theorem1.2If then If then where εα,q and d(α,q) are as defined in Theorem1.1. In particular, for any integer1≤q <X/4, we haveTheorem1.3If then If then where ε*/α,q=1or0according to if there exists a positive integer q satisfying||α|-(?)q|≤X-1/2with1≤|α|<(?)X/4,and In particular, for integer1≤q <X/16, one has Using the same method,We are able to improve the result in[26]for holomorphic forms.We state the result as followsTheorem1.4Let λf(n)be the n-th Fourier coefficient of a holomorphic cusp form f of weight k for SL2(Z).If then If then for β≠1/2,In Theorem1.4we improve the upper bound from Xmax{β,1/2-β/4}+ε in [26]to x1/3+εFor the second part in this thesis,we consider the weighted exponential sums twisted by Fourier coefficients of Maass cusp forms for SL(3,Z) whereΦ(x)be a C∞function on R,which is compact supported on[1/4,5/4]上, identically one on [1/2,1],and has derivatives satisfying: For this question,Ren and Ye([28])proved when max{2max{β,1/3}αβ,1}m1/3≤X1/3-β. the implied constant depend on f,A and β. Using the method of ([28]) together with Pitt ([25]), we have the following theorem.Theorem1.5Let f be α Maass form for SL(3, Z). Af(m,n) is the Fourier coefficient of f, X>9.If then for arbitrary big A and small ε>0, where the implied constant depend on f, A and ε only.The tools used in the proof of Theorem1.5are the expansion of δ, Voronoi’s summation formulas and so on.We will introduce the history of the exponential sums with Fourier coeffi-cient of Maass form for SL(2, Z) and SL(3, Z) in section1, and give out the main results. Then we will give proof in section2and section3respectively.