Integral Estimation Of Automorphic L-function On GL₂

Many scholars have done a lot of research on the high-order integral mean prob-lem of Riemann-Zeta function and the zero density estimation and obtained good results.In this paper,we will apply their similar methods to study the integration of L-function on GL2 and the zero density estimation.Let H denote the complex upper half plane and ?=SL2(Z)be the full modular group.f be a Maass cusp form with Laplace eigenvalue 1/4+v2,and it is an eigenfunction of all the Hecke operators T(n)as well as the reflection operator T(-1):z?-z.We have T(n)f=?(n)f for n?N,and ?(n)=1.The standard L-function of f is defined asThe series above converge absolutely for Rs>1(see[9]).We define ?f(p)and ?f(p)by?f(p)?f(p)=1,?f(p)+?t(p)=?f(p).So L(s,f)can be written asAbout ?f(p)and ?f(p),the Generalized Ramanujan Conjecture predicts that([8])|?f(p)|=|?f(p)|=1.About this conjecture we have([12],[13])|?p|<<p7/64,|?p|<<p7/64,so we can get that ?f(n)<<n7/64 d(n),where d(n)is for the divisor function.Applying the theorem of Chandrasekharan and Narasimhan[28],we can get thatThe mean value of ?f(n)has been studied by many authors,for example,see[2,4,5,14,17,18,21,22,23,26,27,28].Power moments for L-function on the critical line play a significant role in analytic number theory.Ivic[8]and Liu,Li and Zhang[16],[29]studied the supremum of all numbers m such that?1T|L(?+it,f)mdt<<T1+?,and obtain asymptotic formulas for the second,fourth and sixth powers of L(s,f).Here we are interested in another kind of power moments of L(s,f).Let f be a normalized Maass cusp form for SL2(Z).For any fixed number A>2,we define M(A)as the infimum of all numbers M(?1)such that?1T|L(1/2+it,f)|A dt<<TM+?for any ?>0,where L(s,f)is the automorphic L-function attached to f.The definition is similar to the one made for power moments of ?(s).Naturally we seek upper bounds for M(A).In this paper,we study the results of the upper bounds for M(A)associated with Maass cusp forms for 2?A?6 and obtain the zero density estimates for L(s,f)as its applications.For ?(s)we can use the theory of exponential sums(exponent pairs),but there are unknown in the case of L(s,f),so that the bound of M(A)is considerably weaker than the corresponding bounds of ?(s).Theorem 1.1.If A?2 is a fixed number and M(A)is defined by ?1T|L(1/2+it,f)|A dt<<TM+?,then M(A)?1+A-2/4,2?A<6.Power moments of L-functions have a large number of applications in many branches of analytic number theory.For example,we can use them to get the zero density estimates of L-functions.Define N(?,T)=#{?=?+i?:L(p,f)=0,?-<?<1,0???T},where 0 ??<1.For the automorphic L-function L(s,f)attached to holomorphic cusp form f for SL2(Z),Ivic[8]and Kamiya[11]showed that N(?,T)<<T4(1-?)/3-2?+?,1/2??<1;N(?,T)<<T2(1-?)/?+?,3/4??<1.Later,the first one was proved for Maass forms for SL2(Z)by Sankaranarayanan and Sengupta[19].Recently the second one was proved for Maass forms for SL2(Z)by Xu[24]and Tang[20].Also[25]and[3]studied the zero density estimates for automorphic L-functions of GL(m).In this paper,we use the higher power moments of L-functions to improve the previous result.Theorem 1.2.If A?2 is a fixed number,thenthe first bound is valid when 5/2<A?6,and the second one is valid when 4<A?6.Corollary 1.3.If we take A=6 in Theorem 1.2,thenNote 1.4.We note that the first bound is better than T4(1-?)/3-2?+? for 5/6<?<1.Theorem 1.5.If A>2 is a fixed number,then N(?,T)<<T2-1?/?+??3/4??<1,when 2?A?6.