| An L-function is a generating function defined by a Dirichlet series with an Euler product. They might be formed out of local data with either an arithmetic-geometric object, such as an elliptic curve denned over a number field, or an automorphic form. The conjectures in the Langlands program pointed out that, despite their different classical origins, every general L-function should in fact factorize into a product of L-functions attached to the class of cuspidal automorphic representations on some GLm.In this article, what we are concerned are these fundamental L-functions stated in Langlands' observation, i.e., the automorphic L-function L(s,π) attached to an irreducible unitary cuspidal representationπof GLm(AQ) with m≥2.In Chapter 1, we begin by introducing the brief history of L-functions, and the main features in the theory of L(s,π). To be general, we give the definition of L(s,π) in a representation-theoretical way, and give the case of primitive forms as an example.Our main results are the prime number theorems for L(s,π) under the Generalized Riemann Hypothesis (GRH in brief), which are stated in Chapter 2. The remaining chapters in this thesis are the detailed proofs.We link L(s,π) with primes by defining the counting functionIn this way, the Prime Number Theorem for L(s,π) studies of the asymptotic behavior ofψ(x,π).The known unconditional estimation isψ(x,π)(?) x cxp {-c(?)} for some positive constant c, with the implied constant depending onπ. This is a special case of the main theorem in Liu and Ye [28], and also follows from Theorem 5.13 in Iwaniec and Kowalski [14].The GRH predicts that all the non-trivial zeros of L(s,π) lie on the critical line Rs = 1/2.Under GRH, (0.0.3) is influenced toψ(x,π) (?) x1/2 log2 x (0.0.4)with the implied constant depending onπ.Our theorems not only improve on it, but also give the integral mean value estimates for the prime counting functionψ(x,π) and the normal density estimate in short intervals.Theorem 2.1. Letπbe an irreducible unitary cuspidal representation of GLm(AQ) with m≥2. Assume GRH for L(s,π). We haveψ(x,π) (?)x1/2(loglogx)2for x≥2, except on a set E of finite logarithmic measure, i.e.Theorem 2.2. Letπbe as in Theorem 2.1, and assume GRH for L(s,π). ThenTheorem 2.3. Letπbe as in Theorem 2.1, and assume GRH for L(s,π). We have where C > 0 is a constant depending onπ. By denoting the upper bound 6 towards the Generalized Ramanujan conjecture, we also get the following.Theorem 2.4. Letπbe as in Theorem 2.1, and letθbe as in (A8). Assume GRH for L(s,π). We havefor any increasing functions h(x)≤x satisfying h(x)/(xθlog2x)→∞.Our Theorems 2.1-2.4 generalized the classical results, as for the caseζ(s) in Gallagher [8], Cramer [2] and [3], and Selberg [38].Our proofs combine Gallagher's approach in [8] and [9], with recent results of Liu and Ye [26] [28] and Liu, Wang, and Ye [27] on the prime number theorem for Rankin-Selberg automorphic L-functions. The main improvement in the proof is due to an explicit formula.Theorem 3.1. Letπbe an irreducible unitary cuspidal representation of GLm(Aq) with m≥2. Then, for x≥2 and T≥2,whereθis the upper bound towards the generalized Ramanujan conjecture.The significance of Theorem 3.1 is that it is unconditional. Explicit formulae of different forms were established by Moreno [30] [31]; under GRC, explicit formulae for general L-functions were proved in (5.53) of Iwaniec and Kowalski [14]. Our explicit formula requires neither GRH nor GRC, such that it could be used for general problems relating to automorphic L-functions L(s,π).
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