Power Moments Of Automorphic L-function Attached To Maass Forms

Let Γ=SL2(Z) be the full modular group and H denote the complex upper half plane. The Laplace operator has a spectral decomposition on L2(Γ\H), which states that Here C is the space of constant functions, C(Γ\H) is the space of Maass cusp forms and ε(Γ\H) is the space spanned by the incomplete Eisenstein series.Let U={uj:j≥1} be an orthonormal basis of Hecke-Maass forms with Laplacian eigenvalues1/4+tj2with tj≥0in the space C(Γ\H). Then uj has the Fourier expansion where ρj(1)≠0,λj(n) is the eigenvalue of n-th Hecke operator Tn, Ws(z) is the Whittaker function given by where e{x)=e2πix and Ks(y) is the K-Bessel function with s=1/2+it.For convenience, throughout this paper, let f be a Maass form with Laplace eigenvalue1/4+v2. We normalize f with the first coefficient being1, so the Fourier expansion of f becomes Then the L-function attached to f is defined as The series above converge absolutely for Re s>1(see [6]). Also we note that L(s, f) satisfies a functional equation and can be analytically continued to the whole complex plane (see [1]). Applying the Theorem of K. Chandrasekharan and R. Narasimhan [2], we can deduce that By this bound and Cauchy’s inequality, we have which makes us can use Lemma2.5bellow. We define αf(p) and βf(p) by Then L(s,f) can be written as The Generalized Ramanujan Conjecture predicts that (see [8]) About this conjecture the best known result is due to Kim and Sarnak [10],[11]: consequently we have λf(n)<<n1/64d(n), where d(n)=Σd|n1is the divisor function.For1/2<σ<1fixed, we define m(σ)(≥2) as the supremum of all numbers m (>2) such thatwhere the <<-constants may depend on L(s,f) and ε. The definition is analogous to the one made for power moments of ξ(s).Naturally we seek lower bounds for m(σ), which has some applications for the mean value estimate of Fourier coefficients of f. In1989, A. Ivic [4] studied a similar problem for holomorphic cusp forms and obtained a lower bound2/3-4σ for1/2<σ≤5/8.In this paper, we want to show the results of m(σ) associated with Maass cusp forms in the interval (1/2,1). For1/2<σ≤5/8, we shall get the same bound as holomorphic cusp forms, the difference is that for Mass cusp forms we do not know whether the Ramanujan Conjecture is true, which causes difficulties to establish bounds like lemmas2.11and2.12. When5/8<σ≤1-ε, the bound of m(σ) is considerably weaker than the corresponding bounds of ξ(s)-Because for ξ(s) we can use the theory of exponential sums (exponent pairs), which are absent in the case of L(s,f).Theorem1.1Let m(a) be defined by (1.1), then for1/2<σ<1,we have As an application of Theorem1.1, we shall also give asymptotic formula for the second, fourth and sixth power of L(s,f).Theorem1.2For any ε>0and a fixed satisfying1/2<σ<1we havewhere is the Dirichlet convolution of λf with itself;(1.2),(1.3) and (1.4) follows for1/2<σ<1,5/8<σ<1and respectively.