The Generalized Prime Number Theorem For Automorphic L-Functions

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(QA). 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 the automorphic L-functions.In this thesis, we study the generalized prime number theorem for automorphic L-functions attached to the automorphic representations for GLm(QA) and GLm(EA).Letπbe an automorphic irreducible cuspidal representation of GLm(QA) with unitary characters, and L(s,π) be the corresponding automorphic L-function which is given by products for local factors for SRs＞1 as in Godement and Jacquet [11]: where Here aπ(p,j) are complex numbers associated withπp according to the Langlands correspondence.To link L(s,π) with primes, we take logarithmic differentiation of it, then for SRS＞1, we have whereΛ(n) is the von Mangoldt function, and The prime number theorem for L(s,π) was considered by Qu [38] under Generalized Riemann Hypothesis, and she obtained that except on a set of x of finite logarithmic measure.Ifπ′is an automorphic irreducible cuspidal representation of GLm′(QA), we define L(s,π′),απ′(p, i),and aπ′(pk) likewise, for i= 1,..., m′. Ifπandπ′are equivalent, then m=m′and {απ(p, j)}={απ′(p,i)} for any p. Hence aπ(n)= aπ′(n) for any n=pk, whenπ≌=π′.Forπandπ′as above, one can obtain the corresponding Rankin-Selberg L-functions L(s,π×π′) which are developed by Jacquet, Piatetski-Shapiro, and Shalika [20], Shahidi [43]. This L-function is given by local factors: where Similarly, we haveLiu and Ye [31] considered the prime number theorem of Rankin-Selberg L-functions which concerns the asymptotic behavior of the function The main theorem asserted that under the condition that at least one ofπandπ′is self-contragredient.In Chapter 1 of this dissertation, we investigate the generalized prime number theorem of Rankin-Selberg L-functions L(s,π×π). Consider where k is a positive integer, andρπ×π(n) is a complex number attached toπandπ.Theorem 1.1. Letπbe an automorphic irreducible cuspidal representation of GLm(QA). Assume thatπis self-contragredient:π≌π. Then where the complex constants a_j,k (j=1,..., k-1) are attached toπandπ.Let E be a Galois extension of Q of degree l. Let EA=Π′vEv be its adele ring, where v runs over all places of E, andΠ′denotes a restricted product. For any prime p, we have E(?)Qp=(?)v|pEv, where v with v|p are places of E lying above p. Since E is Galois over Q, all Ev with v|p are isomorphic. Denote by lp the degree, by ep=ordv(p) the order of ramification, and by fp the modular degree of Ev over Q for v|p. Then lp=epfp and qv=pfp is the module of Ev. On the other hand,E(?)QR is either(?)v|∞R or (?)v|∞C.Let (?) and (?)′be automorphic irreducible cuspidal representations of GLm(EA) and GLm′(EA) with unitary central character, respectively. Denote L(s,(?)×(?)) the corresponding Rankin-Selberg L-functions. Consider Recently, Gillespie and Ji [9] studied the sumΣn≤xΛ(n)a(?)×(?)′(n) and asserted that under the condition that at least one of (?) and (?)′is self-contragredient.In Chapter 2 of this dissertation, we consider where k is a positive integer, andρ(?)×(?)(n) is a complex number attached to (?) and (?).Theorem 2.1. Let (?) be an automorphic irreducible cuspidal representation of GLm(EA). Assume that (?) is self-contragredient:(?)≌(?). Then where the complex constants C_j,k (j=1,..., k-1) are attached to (?) and (?).In 2009, Lu [29] studied the sum∑n≤xλπ(n) and obtained that whereπis unramified at every finite place p, andεis an arbitrarily small number.In Chapter 3 of this dissertation, we are actually studying the estimate of the sumΣn≤xλ(?)(n).Theorem 3.1. Let (?) be an automorphic irreducible cuspidal representation of GLm(EA). Denote by L(s, (?)) the automorphic L-function attached to (?), andλ(?)(n) the corresponding n-th coefficient. If (?) is unramified at every finite place p, then we have for anyε＞0...