Zero-Density Estimates Of L-Functions For Maass Cusp Forms

The zero-density estimates for L-functions is a significant topic in ana-lytic number theory.Many mathematicians researched on this problem and proved many significant results.In this thesis we will also study this problem.Following the classical line of Halas-Montgomery-Jutila-Ivic,and introducing some different techniques,we get a result about the zero-density estimates.Our result is a little sharper than the previous results when o,is close to 1.Let/be a normalized Maass cusp form for the full modular group SL2(Z)which is an eigenfunction of all the Hecke operators T(n).We have T(n)(f)=?(n)f for n ? N,and ?(1)= 1.Associated to/,there is an L-function L(s,f),which is defined asL(s,f)=(?)?f(n)/ns=?(1-?f(p)p-s +p-2s)-1,Re(s)>1.The above series converges absolutely for Re(s)>1,and the function L(s,f)can be continued analytically to the whole complex plane.Similar to the classical Riemann zeta-function or Dirichlet L-function,all the non-trivial zeros of L(s,f)are in the area 0<Re(s)<1.It is conjectured that the non-trivial zeros of L(s,f)lie on the critical line Re(s)= 1/2(GRH for short).The GRH is very difficult to prove.A weaker conjecture is the zero-density hypothesis.LetNf(?,T):=#{?=?+i?:L(p,f)= 0,???,|?| ? T}.Then the zero-density hypothesis states that the estimate Nf(?,T)<<T2(1-?)+?holds for ?>1/2 and A(?)= 2.This estimate is still open so far.For L(s,f)being the L-function attached to a holomorphic cusp form on SL2(Z),Ivic has shown in[3]thatNf(?,T)<<T2(1-?)/?+? 3/4???1.Recently,H.Tang[9]proved that the above result also holds for Maass cusp forms for SL2(Z).It is well known that if f is a holomorphic cusp form,then ?f(n)?R(see[4])and |?f(n)|<<n?(see[1]).While for a Maass cusp form,the famous Ramanujan Conjecture |?f(n)|<<n? has not been proved,and the best known result towards this conjecture is due to Kim and Sanark[5,6]|?f(n)|<<n7/64+?.This will bring some difficulties in following the classical line of zero-density estimates,especially in applying the Halas-Montgomery inequality(see Lem-ma 2.2).However,we can use an average estimate of the(?f(n))4 to deal with these difficulties(see Lemma 3.1).Similar techniques can also be seen in A.Sankaranarayanan and J.Sengupta[8],H.Tang[9],Z.Xu[10],Y.Ye and D.Zhang[12].This technique implies that the Ramanujan Conjecture is true on average.This is enough for us to apply the classical approaches and to get the following new result about the zero-density estimates of L(s,f):(?)...