| It is an important problem in analytic number theory to study the moments of Riemannζ-function and L-functions. These theories play an important role in number theory. In this thesis, we will study the moments of automorphic L-functions associated to holomorphic cusp form.To state our results we need some notations. Letκbe an even positive integer and f(z) be a holomorphic cusp form of weightκwith respect to the full modular group SL2(Z). Moreover, we assume that f(z) is a normalized eigenfunction for all Hecke operators. In this case f(z) has the following Fourier series expansion Such an f is called a holomorphic Hecke eigenform. Associated to each Hecke eigenform f, there is an L-function L(f, s), which is defined as for Res> 1.In first two sections of this paper, we will consider the power moments of the integral of L(f, s), i.e. where k> 0 is a real number. For Riemann zeta-function, many authors (see [1], [3], [15] etc.) considered the moments and conjectured that Mk(T)～CkT(logT)k2 for a positive constant Ck. But asymptotic formulae for Mk(T) has been established only for k= 1 (due to Hardy and Littlewood, see [30]) and k= 2 (due to Ingham, see [30]). However, many authors considered the upper and the lower bound of Mk(T). Especially for the lower bound, we do have Mk(T)>>T(logT)k2 in many cases. For example, Heath-Brown [8] proved that for k> 0 is rational, Mk(T)>>T(logT)k2. For k= (?)> 0, where n is an integer, Mk(T)< 0. Then, as T→∞, Assume GRH for L(f, s) is true, then last inequality is ture for any real number k> 0, and for any real number 0< k< 1 we haveWhen k is large, it seems difficult to get the conjectured upper bound for Mk(T) and Mk(f,T). Recently, Soundararajan [29] gave another method, and proved that, under RH, for every positive real number k and every (?)> 0, Above bound is very close to the conjectured bound, although it does not seem possible to remove e by the present method. However the range of k can be enlarged to all the positive real numbers. Using the same method, Ivic [10] obtained a short interval result, under RH, for every positive real number k, where H = Tθwith fixed 0 <θ≤1.In the second section, using the method in [29] and some results of L(f, s), we will study the upper bound for Mk(f,T). We will prove the following results.Theorem 2.1. Letλ0 denote the unique positive real number satisfying (?). Assume GRH for L(f, s), for any real number k > 2e(?), we haveTheorem 2.2. Letλ0 denote the unique positive real number satisfying (?). Assume GRH for L(f, s), for any real number k > 2e(?) , we have where H = Tθwith 0<θ≤1.Many authors have also considered the sum of power of L-functions, see [25], [26], [28] etc.. Recently, the moments of twisting L-functions has received much attention. In [28], Soundararajan and Young considered the second moment of twisting L-functions L(s,f(?)χd), and gave the lower bound. In chapter 3, we will consider the moment of another twisting L-functions, i.e. where L(s,f(?)χ) is the L-function associated to the twist of cusp form f by Dirichlet characterχ, defined as following, We will use the method of [9] and [25] to estimate the lower bound and upper bound of Mk(q,f) respectively. Heath-Brown [9] and Rudnick and Soundararajan [25] studied the sum of power of Diriechlet L-functions respectively, which is where q is natural number andχis the Dirichlet character modulo q.Using the method of [9], we will consider the upper bound of Mk(q,f). Our result isTheorem 3.1. Under GRH for L(s, f(?)χ), for any real number 0< k< 1 we haveFor the the lower bound of Mk(q,f), we will use the method of [25]. Our result isTheorem 3.2. Under GRH for L(s, f(?)χ) for any natural number k and any natural number q we have...
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