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1、Pure and Applied Mathematics Quarterly Volume 3, Number 2 (Special Issue: In honor of Leon Simon, Part 1 of 2) 481497, 2007Perrons Formula and the Prime Number Theorem for Automorphic L-FunctionsJianya Liu1and Yangbo Ye2Abstract: In this paper the classical Perrons formula is modified so that it now
2、 depends no longer on sizes of individual terms but on a sum over a short interval. When applied to automorphic L-functions, this new Per- rons formula may allow one to avoid estimation of individual Fourier coef-ficients, without assuming the Generalized Ramanujan Conjecture (GRC). As an applicatio
3、n, a prime number theorem for Rankin-Selberg L-functions L(s, 0) is proved unconditionally without assuming GRC, where and 0are automorphic irreducible cuspidal representations of GLm(QA) and GLm0(QA), respectively. 2000 Mathematics Subject Classification: 11F70, 11M26, 11M41.1. IntroductionThe clas
4、sical Perrons formula gives a formula for a sum of complex numbers an, 1 n x, in terms of their Dirichlet seriesf(s) =Xn=1an ns(1.1)Received November 10, 2005. 1Supported in part by China NNSF Grant # 10531060, and by a Ministry of EducationMajor Grant Program in Sciences and Technology. 2Supported
5、in part by the USA National Security Agency under Grant Numbers MDA904-03-1-0066 and H98230-06-1-0075, and by a University of Iowa Mathematical and Physical Sciences Funding Program Award. The United States Government is authorized to reproduce and dis- tribute reprints notwithstanding any copyright
6、 notation herein.482Jianya Liu and Yangbo Yeand bounds for individual terms an, where here and throughout s = + it C. Let A(x) 0 be non-decreasing such that an A(n), and letB() =Xn=1|an| n(1.2)for a, the abscissa of absolute convergence of (1.1).Then the classical Perrons formula (see e.g. Heath-Bro
7、wns notes on Titchmarsh 31, p.70) states that, for b a, Xnxan=1 2iZb+iTbiTf(s)xssds + O?A(2x)xlogxT+O?xbB(b)T + O A(N)min?x T|x N|,1? ,(1.3)where N is the integer nearest to x.When applying (1.3) to the Riemann zeta-function or Dirichlet L-functions, bounds for anpose no problem.When applying this f
8、ormula to other auto- morphic L-functions, however, bounds for anoften require an assumption of the Generalized Ramanujan Conjecture (GRC). Examples include a prime number theorem for Rankin-Selberg L-functions (Theorem 2.3 below) recently proved by the authors in 18 under the GRC.In this paper, we
9、will prove a revised version of Perrons formula (Theorem2.1 and Corollary 2.2 below). Different from the classical (1.3), the new Perrons formula produces a formula forP nxanin terms of a sum of |an| over a shortinterval. While bounding individual Fourier coefficients |a(n)| of an automor- phic cusp
10、idal representation is hard and may require GRC, estimation of a sum of |a(n)| can usually be done by the Rankin-Selberg method. The new Per- rons formula thus allows us to prove certain results for automorphic L-functions without assuming the GRC.As an application, we are now able to prove a prime
11、number theorem (Theorem 2.3) unconditionally for Rankin-Selberg L-functions L(s, 0), by removing the assumption of GRC in 18. This prime number theorem has a remainder termof a size which reflects our current knowledge of zero-free regions of L(s,) as in (4.3) and (4.4). We will see that the new Per
12、rons formula allows us to deduce the prime number theorem for 6= 0from the diagonal case of = 0.Several authors have already addressed the question of prime number theorem for Rankin-Selberg L-functions in the GL2context, and they all faced the problemof bounding Fourier coefficients.Moreno 20 avoid
13、ed GRC by an averaging technique, while others restricted themselves to the case of holomorphic cusp forms where GRC is known (Ichihara 5), or to the Selberg class where GRC is assumed (Kaczorowski and Perelli 11).Perrons formula for automorphic L-functions483We remark that using these prime number
14、theorems one can count primesweighted by Fourier coefficients of automorphic cuspidal representations. This can be regarded as a direct connection between representation theory and prime distribution.2. Main theoremsThe following is a modification of (1.3).Theorem 2.1. Let f(s) be as in (1.1) and ab
15、solutely convergent for a. Let B() be as in (1.2). Then, for b a,x 2, T 2, and H 2,Xnxan=1 2iZb+iTbiTf(s)xssds + OXxx/H a= 1,B() =Xn=1(n) n1 1.Therefore, Corollary 2.2 with b = 1 + 1/logx givesXnx(n) =1 2iZb+iTbiT 0(s)(s)?xs sds + Oxlogx T? .484Jianya Liu and Yangbo YeWe can take T = exp(logx). The
16、prime number theorem Xnx(n) = x + Oxexp(cp logx)now follows from the zero-free region of the Riemann zeta-function and a standard contour-integration argument; here and throughout c denotes a positive constantnot necessarily the same at different occurrences.In order to describe applications of this new Perrons formula to automorphic L-functions, let us recall that for a
