The First Moment Of Rankin-Selberg L-functions At Special Points

In 1837,Dirichlet first introduced Dirichlet L-functions in order to solve difficult problems in number theory,which means the beginning of analytic number theory.This research technique has been developed greatly in the last century and has become an important method for studying modern number theory.Now,the term L-functions is no longer limited to ?(s)and Dirichlet L-functions,other objects like modular forms and automorphic forms also correspond to their own L-functions which motivate the study for modular forms and automorphic forms greatly.Rankin-Selberg method was invented by Rankin and Selberg individually in 1939 and 1940[1][2]and was used to study certain L-functions related to the Fourier coefficients of two modular forms,later,this method was extended to the study of Maass forms.We will discuss a family of selected Rankin-Selberg L-functions in this article.The analytic behaviour of L-functions on 0<Rs<1 is always a central topic in number theory.In 1993,Wenzhi Luo[3]proved a first moment formula on special points sj=1/2+itj for a series of L-functions,by which he deduced that when T is big enough,we have following asymptotic formula:here Q is a normalized Hecke eigenform of weight 4,{uj} are the orthogonal bases of Hecke-Maass forms,L(s,Q × uj)is the Rankin-Selberg L-function induced by Q and uj and tj is the spectrum parameters for uj(see chapter 2.1 for further details).When tj is much bigger than T,the rapidly decreasing of e-tj/T makes this term negligible,thus this summation is taken essentially on[0,T1+?].In analytic number theory,results like this are called results on long intervals.In addition to Rs=1/2 the behaviour of Rankin-Selberg L-functions on Rs = 1 also have important meaning,for example,Goldfeld-Hoffstein-Lieman[4]proved that which indicates that L(1,sym2(g))has no real zeros in(1-c/log tg,1),here c is a constant,4 is a Maass form on GL(2)with tg its spectrum parameter.This thesis is to investigate the first moment of those GL(2)x GL(2)Rankin-Selberg L-functions at special points on Rs = 1,and we get following result on short intervals:with T1/2+?? M ? T1-?,f a cusp Hecke modular form,{uj} the orthogonal basis of Hecke-Maass forms,tj the spectrum parameter for uj.It's easy to see that,due to the rapidly decreasing of e-(t-T)2/M2,this summation is essentially taken on[T-M1+?,T+M1+?]which is pretty short,compared with T1+?.By applying approximate functional equation and Kuznetsov trace for-mula,we can get the major terms from the summation of diagonal terms and thus convert this problem to the estimation for the summations of continuous spectrum terms and non-diagonal terms.Since L(s,f × uj)behave well on points 1 + itj,we can change the intervals for integration and summations gradually by adding restrictions on M.Finally,we end the proof by choosing adequate M to make the main term larger than all minor terms in our proof.We will refer to the method used by Xiaoqing Li[5].