| 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 defined over a number field) or with an automorphic form. According to conjec-tures in the Langlands Program, any "most general" L-function should be a product of L-functions of automorphic cuspidal representations of GLm/Q. Other parts of the Langlands conjectures imply that the Ramanujan-Petersson conjecture should hold for any automorphic L-function. Thus it is very important and essential to investigate L-functions.In this paper, we shall consider two kinds of L-functions, the spinor zeta functions and the symmetric square L-functions. We will study the Riesz mean values of their coefficients. Before stating our results we need some definitions.Let{cn} be a sequence of complex numbers. We define the general Dirichlet series by The Ricsz mean of the coefficients cn of Z(s) is defined by where??0, and?' means that if?=0 and x is an integer, then cx is replaced by cx/2. Rankin [34] and Selberg [37] independently introduced the following function where?(s) is the the Riemann zeta-function, a(n) is the n-th Fourier coefficient of the holomorphic cusp form f(z) for the full modular group SL2(Z). In the half plane?=Rs>1, the function Z(s) has the absolutely convergent Dirichlet series expansion with Dclignc's estimate|a(n)|<<n(?-1)/2?(n) (cf [6]) implies cn=O(n(?)). Here and in what follows (?) denotes an arbitrarily small positive number which is not necessarily the same at each occurrence.In [16], Ivic, Matsumoto and Tanigawa studied the Riesz mean of the type where?2?R0/6 is the residue of Z(s) at the simple pole s=1. Then they established the Voronoi-type formula for the error term??(x;f).Recently, Ivic [18] studied the 4-th power moment of?1(x;f) and obtained an upper bound. In [40], the authors got an asymptotic formula, which substantially improved Ivic's estimate. More generally, they established the asymptotic formula for the k-th power moment of?1(x;f) with k=3,4,5. Following similar approaches in [16,18,40], we shall study the Riesz mean values of the spinor zeta functions and the symmetric square L-functions, and finally give the estimates for the higher power moments.In Chapter 1, we begin by introducing the spinor zeta functions. Let S?(?g) be the space of cusp forms of weight?and genus g on the full Siegel modular group?g. For F?S?(?g), we suppose F is a common Hecke eigenfunction, i.e. We then define where?1,p,?2,p,…,?g,p arc the Satake p-paramctcrs. For R>>0, define the spinor zeta function byIn this thesis, we consider the case g = 2. In order that ZF(s) has nice analytic properties we suppose that F satisfies one of the following conditions1.?is odd;2.?is even and F?S?*(?2)?, where S?*(?2) is a Maass subspace of S?(?2) and is defined in Lemma 1.2. By Lemma 1.3, ZF(s) is entire and in particular vanishes at the points s=?-2,?-3,…. On the other hand, under our assumption on F, by Weissaucr's theorem (cf [43] Chap.1) the Satake p-paramctcrs in (0.0.2) have absolute value 1. Then we obtain We conclude that ZF(s) converges absolutely for Rs>k-1/2 and so ZF(s)?0 in this range.When ZF(s) is entire, we will establish the following Voronoi-type formula.Theorem 1. Let F?S?(?2). Assume that F?S?*(?2)?if?is even. Fix 1/2<p?1. Let N>>1. Then for any x>1,For the mean square of D?(x; F), we will prove the following. Theorem 2. Under conditions of Theorem 1. We have whereIn Chapter 3, we shall consider the higher power moments of D?(x;F) and obtain the following theorems.Theorem 3. Under conditions of Theorem 1, if there exists a real number A0>3 such that then for any integer 3?h?A0, we have the asymptotic formula where B?(h;f) and??(h, Ao) are defined in (1.3.3) and (1.3.4).Theorem 4. For?=1, (0.0.3) holds with A0=16/3. Thus the asymptotic formula holds for h=3,4,5, and??(h, A0) is given by (1.3.5).We must point out that, for 1/2<?<1, the method used to prove Theorem 4 is failed in finding a number A0 which satisfies (0.0.3).We will consider the symmetric square L-functions in Chapter 4. Let f(z) be a holomorphic cusp form of weight?with respect to the full modular group SL2(Z) Fomenko [10,11] considered the Ricsz mean problem on the following symmetric square L-functions Write cn=?sym2f(n). Then it is known that for any (?)>0, and?3(n) is the number of ways to write n as a product of 3 factors.Define the error term??(x;sym2f) by In [10,11], Fomenko established the Voronoi-type formula of?p(x;sym2f) and its mean square estimates. Motivated by Fomenko's resultcs, we give some further results in Chapter 4. Precisely, we will proveTheorem 5.If there exists a real number E0:=E0(?)>3 such that holds for any (?)>0, then for any integer 3?k?E0, we have the asymptotic formula where C?(k,c) and??*(k,E0) are given by (1.3.9) and (1.3.11) respectively. Particularly, for?=1/2, we can prove the following unconditionally result.Theorem 6. Let?=1/2. For k=3,4,5;we have where??*(k, Eo) is given by (1.3.12).In order to prove Theorem 1, we will use the complex analytical method and the convexity bound for ZF(s) in a critical strip. For details, see Section 2.1 and 2.2. About the proof of Theorem 2. It suffices for us to take square of Dp(x;F), and then estimate the integral term by term. For the proof of Theorem 3. Let R1 denote the partial sum of n?y (y>T(?)) in the Voronoi formula of R1. The sum of the remainder is R2. We will prove that the higher power moments of R2 is very small. Therefore we can estimate?1T D?h(x;F)dx well by the bound of?1T R1hdx. In the proof of Theorem 4, we will use the large value estimate Theorem, Kuzmin-Landau inequality, the exponent pair, etc.. The proof of Theorem 5 and Theorem 6 are similar as Theorem 3 and Theorem 4 respectively. |
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