The Average Estimates For Coefficients Of Automorphic L-functions

Generally speaking, an L-function is a type of generating function formed out of local data associated with either an arithmetic-geometric object ( such as an elliptic curve denned over a number field ) or with an automorphic form. Accordingto conjectures in the Langlands Program, any "most general" L-function should be a product of L-functions of automorphic cuspidal representations of GL_m/Q. Other parts of the Langlands conjectures imply that the Ramanujan-Peterssonconjecture should hold for any automorphic L-function. Thus it is very important and essential to investigate the automorphic L-functions.In this thesis, we study the average behavior of coefficients of automorphic L-functionsattached to holomorphic cusp forms for the full modular group SL₂(Z).Let k be a positive even integer, and H_k^* be the set of all normalized Hecke primitive eigencuspforms of weight k for SL₂(Z). The Fourier expansion of f∈H_k^* at the cusp∞iswhereλ_f(n) is the eigenvalue of the (normalized) Hecke operator T_n. Thenλ_f(n) is real and satisfies the multiplicative propertywhere m≥1 and n≥1 are any integers. In 1974, P. Deligne [3] proved the Ramanujan-Petersson conjecturewhere d(n) is the divisor function. The Hecke L-function attached to∫∈H_k^* is defined byIn Chapter 1 of this dissertation, we investigate the average behavior of coefficients of Hecke L-functions over sparse sequences by using the properties of symmetric power L-functions and their Rankin-Selberg L-functions, which have been established in [6], [11], [12], [13], [14], and [23],Rankin [17] and Selberg [22] invented the powerful Rankin-Selberg method to study the average behavior ofλ_f²(n), and showed thatFirst of all, in Chapter 1 we obtain the asymptotic formula for the sumTheorem 1.1. Let f∈H_k^*, andλ_f(n) denote its n-th normalized Fourier coefficient. Then for anyε＞0, we havewhere j =2,3,4.In 1983, Moreno and Shahidi [24] were able to provewhereÏ„₀(n)=Ï„(n)/n^11/2 is the normalized Ramanujan tau-function. Obviously Moreno and Shahidi's result also holds true if we replaceÏ„₀(n) by the normalized Fourier coefficientλ_f(n).In Chapter 1 we are also interested in the fourth moment of the normalized Fourier coefficient of f∈H_k^* over square numbers, i.e. Theorem 1.2. Let∫∈H_k^*, andλ_f(n) denote its n-th normalized Fourier coefficient. Then for anyε＞0, we havewhere P₂(t) is a polynomial in t of degree 2.Recently Fomenko [5] studied the mean square estimate for the coefficients of the symmetric square L-function attached to f∈H_k^*, and showed thatIn Chapter 2 we are interested in the integral mean square estimates for the coefficients of the j-th symmetric power L-function with j = 3,4, namelyBy using the properties of symmetric power L-functions and their Rankin-SelbergL-functions, and the mean square formula of the error term for a class of arithmetical functions(See [1] and [16]), we are able to establish the following results.Theorem 2.1. Let∫∈H_k^*, andλ_{sym^j f}(n) denote the coefficients of the j-th symmetric power L-function. Then we haveVinogradov [30] first studiedwhere (?)(n) is the von Mangoldt function. Actually he showed that Let (?)(n) be the coefficient of the logarithmic derivative for the Hecke L-function.In Chapter 3 we study the cancelation of the function (?)(n) twisted with a character e(α(?)),α＞0, i.e.where x＞2. Note thatWe are actually studying the mean value estimate for the coefficients of cusp forms in exponential sums over primes.Theorem 3.1. For anyα＞0 and any sufficiently smallε＞0, we havewhere the implied constant depends onαand the cusp form f.