Some Analytic Results Of Automorphic L-Functions

Let be a generalized L-function where s =σ+it is a complex variable. An important problem in analytic number theory is to estimate the moments (0.1) forσ≥(?) and a fixed positive real number k. On and near the critical lineσ=(?), it is sometimes very difficult to obtain an asymptotic formula for the generalized L-functions with high moments. In particular, it is a long history for estimating the integral moments (0.1) for Riemann zeta-functionζ(s); see [11], [41] for detailed.Letπbe an irreducible cuspidal automorphic representation of GLm(AQ) with the unitary central character and s=σ+it∈C. Let be the standard automorphic L-function, see [7], attached toπ.We give a simple method to obtain lower bounds for integral moments of the standard L-function L(s,π) over short intervals. One of our main theorems is as follows.Theorem 2.1. Letπbe an irreducible cuspidal automorphic representation of GLm(AQ) with the unitary central character and let k be any positive real number. Then uniformly inσ, for all T≥T0 for some sufficiently large T0,σ≥1/2, and T≥H≥log1+(?) T with any (?)> 0.Remarks:For Lindelof hypothesis of Riemann zeta-functionζ(s), is equivalent to, for any integer k and anyσ> 1/2 as T→∞. It can be generalized to automorphic L-functions L(s,π). Therefore, we get the best possible lower bound forσ> 1/2 in Theorem 2.1.Let f(z) be a holomorphic cusp form of weightκwith respect to the full modular group SL2(Z). Let L(s,f) be the automorphic L-function associated with f(z) and letχbe a Dirichlet character modulo q. Forσ> 1, the automorphic L-functions L(s, f(?)χ) are defined byFor any positive real number k, we defineFor the case of Dirichlet L-functions L(s,χ), Heath-Brown [10] considered the upper bound of motivated by Heath-Brown's work [9], which is based on a convexity theorem for mean-value integrals. For our case, we follow the argument of [10] and get the following the results.Theorem 3.1. For 0< k≤1/2, under Generalized Riemann Hypothesis for L(s,f (?)χ), we haveFor two irreducible automorphic cuspidal representationsπandπ' of GLm(AE) and GLm'(AE), respectively, denote the usual Rankin-Selberg L-function by where Forσ> 1, we have see§4.2, for the detailed definition of aπ×π'(n). By the prime number theorem for Rankin-Selberg L-functions L(s,π×π'), we mean the asymptotic behavior of the sumThe prime number theorem for Rankin-Selberg L-functions withπandπ' being classical holomorphic cusp forms has been studied by several authors. Recently, Liu and Ye [24] computed a revised version of Perron's formula. Using the new Perron's formula, the authors proved unconditionally the prime number theorem for Rankin-Selberg L-functions over Q, without assuming the Generalized Ramanujan Conjecture. Following the method in [24], we obtain the prime number theorem for the Rankin-Selberg L-functions defined over a number field E. Theorem 4.1. Let E be Galois extension of Q of degree l. Letπandπ' be irre-ducible unitary cuspidal representations of GLm(AE) and GLm'(AE), respectively. Assume that at least one ofπandπ' is self-contragredient. ThenLet E be a cyclic Galois extension of Q of degree l. Letπbe an irreducible cuspidal representation of GLm(AE) with the unitary central character. Suppose thatπis stable under the action of Gal(E/Q). Thanks to Arthur and Clozel [1],πis the base change lift of exactly l nonequivalent cuspidal representations of GLm(AQ), whereηE/Q is the nontrivial character of AQx/Qx attached to the field extension E according to class field theory. By Langlands [19], we getπis an isobaric sum ofπQ,πQ (?)ηE/Q,…,πQ(?)ηE/Ql-1,which means that, (0.2)Similarly, let F be a cyclic Galois extension of Q of degree q. Letπ' be an irreducible cuspidal representation of GLm'(AF) with the unitary central character. And suppose thatπ' is stable under the action of Gal(F/Q). We also have an isobaric sum (0.3) whereπ'Q is an irreducible cuspidal representation of GLm'(AQ) andψF/Q is the nontrivial character of AQx/Qx attached to the field F. Then we define the Rankin-Selberg L-function over the different number fields E and F by (0.4) where L(s, (?)),0≤i≤l-1,0≤j≤q-1 are the usual Rankin-Selberg L-functions over Q with the unitary central characters. Then forσ> 1, we have where By a prime number theorem for Rankin-Selberg L-functions L(s,π×BCπ') over number fields E and F, we mean the asymptotic behavior of the sumFollowing the methods in Liu and Ye [24], we obtain the prime number theorem over different number fields E and F.Theorem 4.2. Let E and F be two cyclic Galois extensions of Q of degrees l and q with (l,q) = 1, respectively. Letπandπ' be irreducible unitary cuspidal representations of GLm(AE) and GLm'(AF), respectively. Assume that we have base change lifts (0.2) and (0.3), also suppose thatπQ is self contragredient, then...