On The Estimates For Fourier Coefficients Of Cusp Forms In Exponential Sums

In analytic number theory, the Fourier coefficients of cusp forms are fasci-nating and important arithmetic functions that have been studied extensively 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 n∈for any∈>0 sufficiently small. In the case of holomorphic cusp forms of SL2(Z), this was proved by Deligne in 1974 as a consequence of his proof of Weil conjecture. However, other cases are still open. Luckily, in numerous practical applications, one needs bounds not necessinarily for individual Fourier coefficient but for mean values of some kind.In this thesis, we study the estimates for the exponential sum where as usual, e(z)= e2πiz, F(n) is a real function of n, and a(n) are the normalized Fourier coefficients of a cusp form.In Chapter 1 of this dissertation, we are concerned with the oscillatory be-havior of the normalized Fourier coefficients of holomorphic cusp forms of SL2 (Z).Letκbe an even integer. We denote by Sκ(SL2 (Z)) the space of holomor-phic cusp forms of weightκfor SL2(Z). Any f∈Sκ(SL2(Z)) has the Fourier expansion for (?)z>0. We also assume that f is a normalized Hecke eigenform so that a(1)=1 and the normalized Fourier coeffieints agree with the Hecke eigenval-ues. Moreover, under this normalization, a(n) are all real and they satisfy some multiplicative properties. The size and oscillations of the coefficients a(n) are objects of special interest in the literature. Deligne's estimate states that a(n)(?) n∈for any∈>0. As n grows the coefficients a(n) vary greatly in sign, as can be seen in the estimate for any∈> 0, where the implied constant depends only on the cusp form f and∈; see Good .Recently, Pitt studied the oscillatory behavior of a(n) in quadratic ex-ponential sums. Precisely, he showed that for anyα,β∈R, for any∈> 0, where the implied constant depends only on the cusp form f and∈. Following the same approach of Pitt , we shall study the exponential sum whereα,β∈R and 0<θ≤1/2. In Chapter 1, we shall establish estimates for S(X,α,β,θ) which are uniform inβ.Theorem 1.1. Let f be a holomorphic Hecke eigenform and a(n) denote its n-th normalized Fourier coefficient. For anyα,β∈R, 0<θ<1/2 orα,β∈R, |α|≤2,θ=1/2, we have S(X,α,β,θ)(?)X1-(θ/2)＋∈, for any∈>0, where the implied constant depends only onα,β, f and∈.The uniformity inβallows us to derive interesting consequences from The-orem 1.1. Recently, Blomer and Lii studied the cancelation behavior of a(n) in arithmetic progressions. By the argument in Lii [28] (or see Iwaniec , p.123), we obtain our next result for cancalation of a(n) restricted to arithmetic progressions.Theorem 1.2. Let f be a holomorphic Hecke eigenform and a(n) denote its n-th normalized Fourier coefficient. Let 1≤q≤X. For anyα,β∈R, 0<θ<1/2 orα,β∈R,|a|≤2,θ=1/2, we have for any∈>0, where the implied constant depends only onα,θ, f and∈.Using same ideas, we also derive from Pitt's results the following estimates for cancalation of a(n) in quadratic exponential sums restricted to arithmetic progressions.Theorem 1.3. Let f be a holomorphic Hecke eigenform and a(n) denote its n-th normalized Fourier coefficient. Let 1≤q≤X. For anyα,β∈R, we have for any∈>0, where the implied constant depends only on f and∈.We note that in the case ofα=0, the linear form we studied in Chapter 1 takes the form: where a(m) are the normalized Fourier coefficients of a cusp form. Obtaining estimtes for S(X, ,(?)) which are uniform in the parameter (?) has been investigated intensively by many authors and are well understood when a(m) are normalized Fourier coefficients of cusp forms on the upper half plane. Generally, for cusp forms on GLn, n≥2, one seeks the upper bound S(X, (?))< Xθ+∈, uniformly in (?), for any∈>0 and some 1/2<θ<1. However, little results have been proved.The additively twisted sums S(X,(?)) have long been connected to important questions in analytic number theory. An interesting application was recently given by Lao . She showed that the oscillatory properties of Fourier coefficients of holomorphic Hecke eigencuspforms over Beatty sequences can be derived from the estimate S(X,(?))(?)X1/2 log X for any (?)∈R, where the implied constant depends only on the cusp form f; see Wilton . The so called Beatty sequences are of integers defined by Bα+β＝｛[αm＋β] m-1∞, where [x] denotes the greatest integer not exceeding x. Lao showed that for almost all real numbers a>1 and any real numberβ, where a(m) are the normalized Fourier coefficients of a holomorphic Hecke eigen-cuspform f for SL2(Z), and the implied constant depends on the cusp form f and a.In Chapter 2, following Lao , we study the cancelation of Maass cusp form coefficients over Beatty sequences on GLn. By the same method of Banks and Shparlinski and Lao , we shall generalize Lao's result to GLn Maass cusp forms. Let f be a Maass cusp form of type v for SLn(2). Then f has a Fourier-Whittaker expansion of form: where SLn-1=SLn-1(Z), Un-1=Un-1(Z) is the group of (n-1) x (n-1) upper triangular matrices with integer entries and ones on the diagonal, M= diag(m1...mn-2|mn-1|,..., m1,1) and Wj(z, v,φ1,...,1) is the Jacquet-Whittaker function. Assume that f is normalized so that A(1,...,1)=1. We also assume that f is an eigenfunction of all the Hecke operators so that the Fourier coefficients A(m1,.... mn-i) satisfy some multiplicative properties. Define Then we. have the following.Theorem 2.1. Let a>1 be of type 1. Let A (m1,..., mn-1) be the normal-ized Fourier coefficients of a Maass cusp form f on GLn. Assume that for anyε>0 and any (?)∈R. Then for anyε>0,we have T(α,β, X)(?) f,α,∈,m1,...mn-2 Xθ+ε, uniformly inβ∈R.In order to prove Theorem 2.1, the main difficulty is that the Ramanujan conjecture has not been proved for Maass cusp forms on GLn, n≥2. Thanks to Rankin-Selberg theory, we know that the normalized Fourier coefficients behaves like a constant on average. This point helps us overcome the above difficulty.For GL2 Maass cusp forms, Hafner proved that S(X, (?))(?)X1/(2+ε), for any (?)∈R and anyε>0, where the implied constant depends only on the cusp form f andε. Thus we have the following.Corollary 2.1. Let a>1 be of type 1. Let a(m) be the normalized Fourier coefficients of a Maass cusp form f on GL2.Then for anyε>0, we have uniformly inβ∈R.Recently, Miller established the nontrivial bound for anyε>0 and any (?)∈R. Thus we have the following. Corollary 2.2. Let a>1 be of type 1. Let A(m1,m2) be the normalized Fourier coefficients of a Maass cusp form f on GL3. Then for any∈> 0, we have uniformly inβ∈R.