| Let f be a holomorphic Hecke cusp form or Hecke-Maass cusp form on SL2(Z),then f has a Fourier expansion.Let ?f(n)be its normalized Fourier coefficients and define the Hecke L-function L(s,f)L(s,f)=(?)=(?)(1-?(p)p-s+p-2s)-1,R(s)>1.It is well known that L(s,f)has similar analytic properties and function equa-tions as the classical Dirichlet L-function L(s,?).For example,L(s,f)can be continued analytically to the complex plane and its non-trivial zeros are in the critical strip 0<Rs<1.The GRH(Generalized Riemann Hypothesis)for L(s,f)states that all its nontrivial zeros are on the line R(s)=1/2.While the Zero-Density Hypothesis states that Nf(?,T)?T2(1-?)logc T or the form without log T Nf((?,T)?T2(1-?)+?.Here c>0,and ?>0 is a sufficiently small constant.Let Nj(??T)denote the number of zeros of Hecke L-function L(s,f)in the region {????1?|?| ?T}When f is a holomorphic Hecke cusp form for SL2(Z),Ivic considered this problem in[19]and proved that Nf(?,T)<<TA(?)(1-?)+?,wherCombining the method of Karatsuba and Voronin,Yashiro gave another proof for the above results in[50].In 2007,Sankaranarayanan and Sengupta[45]investigated this problem when f is a Maass cusp form for SL2(Z),and proved the estimate Nf(?,T)<<T4(1-?)/3-2?log26 T for ??1/2+1/logT.Later,Xu[48]improved this result when 3/4<?<1 by showing that Nf(?,T)<<T(8?-5)(1-?)/-2?2+6?-3 log57T for 3/4??<1.Further improvement was proved by Tang[47]who showed that there exists a fixed constant c>0 such that Nf(?,T)?T2/?(1-?)logc T for 3/4? ?<1.Tang's result was obtained by using the method of Huxley and following the argument of Jutila[27]and Heath-Brown[9].From these estimates one can see that the best uniform bound for Nf(?,T)in the range 1/2 ?? ?1 is Ts(1-?)/3 logc T.If the difference of T? and logcT is ignored,this result is of the same quality with the result obtained by Ivic in the case of holomorphic Hecke cusp form.In this paper,we will prove the following.This improves the results of Ivic and Tang when 3/4<?<1,respectively.Theorem 1 Let f(z)be a normalized holomorphic Hecke cusp form or Hecke-Maass cusp form for SL2(Z).Then we have Nf(?,T)<<TA(?)(1-?)+?,where#12 and the implied constant depends only on f and ?.We know that the zero density results of classical Dirichlet L-functions have important applications in many important number theory problems.In some problems,it can be used to replace the GRH and give very strong results,especially in the problems concerning the distributions of prime numbers.For example,the prime number theorem in short intervals,the distribution of twin prime numbers,exponential sum over primes and so on.By using the estimates of Nf(?,T),we can consider similar problems related to the cusp form f.In this paper,we are interested in the following exponential sum S(a,?,x)=(?)?f(p)(logp)e(?p?),x<p<2x where ??0 and 0<?<1.This problem has been investigated by many authors.When ?=1/2,for fixed ?=-2,Iwaniec,Luo and Saxnak(see[25])gave conditional heuristics result such that S(?,1/2,x)=cfx3/4+O(x5/8+?),where cf is a constant depending only on f.In 2006,Zhao[54]showed that for ?>0,S(a,1/2,x)?x5/6(log x)21,where the implied constants depend on ? and the cusp form f.This is the first unconditional result for such type.However,there is no similar result for ?<0 in[54].In 2010,Pi and Sun considered this problem for general? ?(0,1)in[42]and proved that for ?? 0 and ??(0,9/32]?[1/2,9/16],there holds the estimate S(?,e,x)?x?(x1+?/2+x3/4+?/6+x1-?/2),where the implied constant depends on ?,?,? and the cusp form f.By using Vaughan's identity,Hou[12]proved that for ??0 and 0<??1/2,there exists an absolute constant c>0 such that S(?,?,x)?(x5/6+x1-?/2)(logx)c,where the implied constant depends on ?,?,c>0 and the cusp form f.Recently,for ?>0 and 0<?<1,Jiang and Lu[26]improved this result by showing thatIn this paper,we borrow the idea in[44]and follow the argument in[42]to prove the following estimate.Theorem 2 Let ??0 be fixed and 0<?<1.Then there exists an absolute constant c ? 28 such that for x sufficiently large in terms of ?,?,there holds S(?,?,x)?(x3/4+?/6+x9+5?/14+x1-?/2)logcx,where the implied constant depends on ?,?,c and f.It is easy to see that this estimate is better than previous results when 1/3<?<1/2 and 1/2<?<3/4.Moreover,when 0<??9/16,we have S(?,?,x)?(x3/4+?/6+x1-?/2)logcx.This refines earlier results and gives a bound depending continuously on 0.In[42],Pi and Sun also consider the following exponential sum?f(p)(log p)e(h?p?).For any ??0,??(0,9/32]and h ?[1,x9/32-?]or ? ?[1/2,9/16]and h ?[1,x9/16-?],they obtained the estimate ?f(p)(logp)e(hap?)?(h1/2x1/2+?/2+h1/6x3/4+?/6+h-1/2x1-?/2)x?,where the implied constant depends on ?,?,? and the cusp form f.One may note that the above result does not cover the range of ??(9/32,1/2)and ? ?(9/16,1).For complements,we will prove the following theorem.Theorem 3 For ??0,? ?(0,9/16]and k ?[1,x9/16?? we have?f(p)(logp)e(k?p?)?(k1/2x1/2+?/2+k1/6x3/4+?/6+k-1/2x1-?/2)x?,where the implied constant depends on ?,?,? and the cusp form f.Moreover,for ??(9/16,1)and k ?[x9/16-?,x1-?],we have?f(p)(log p)e(k?p?)?(k?1(k,x?)x?2(k,x,?)+k1/2x1+?/2)(log x)28,where?1(k,x,?)3/2-?2(k,x,?)=3/2+3?/2-and the implied constant depends on ?,? and the cusp form f.Inspired by the above results,some people also investigated the expo-nential sum over primes when g is a Hecke-Maass cusp form with respect to SL2(Z),i.e.U(?,?,x)=?(P)(log p)e(?p?),where ?g(n)are the normalized Fourier coefficients.For ??0,and 0<??1/2,Hou[12]obtained that there exists an absolute constant c such that U(?,?,x)?(x5/6+x1-?/2)(log x)c,where the implied constant depends on ?,?,c>0 and the cusp form g.For ?>0,Jiang and Lu[26]obtained U(?,?,x)?(x5/6+x2+?/3+x1-?/2)(log x)9/2 for 0<?<1,where the implied constant depends on ?,? and the cusp form g.In this thesis,we will prove the following estimates.Theorem 4 Let ??0 be fixed and 0<?<1-7/64=0.890625.Then there exists an absolute constant c? 28 such that for sufficiently large x,we have U(?,.?,x)?x3/4+?/6+x9+5?/14+x1-?/2)logc x+x7/64+?log x.where the implied constant depends on ?,?,c and the cusp form g.Note that 7/64+??(2+?)/3 for 0<0 ?1-21/128.So it is easy to see that Theorem 4 is better than the previous results for 1/3<?<1/2 and 1/2<??107/128=0.8359375.By Theorem 4 and its proof,we can obtain the following corollaries.Corollary 1 Assume the zero-density estimates for L(s,g)satisfying Ng(?,T)?g,?T2(1-?+?.For fixed ??0 and 0<?<1-7/64=0.890625 and sufficiently large x,we have?g(p)e(?p?)?x7/64+?logx+(x2-?/2+x1+?/2)x?,where the implied constant depends on a,0,e and the cusp form g.Corollary 2 Assume the zero-density estimates for L(s,g)satisfying Ng(?,T)?T2(1-?)+?.Then for fixed ??0 and sufficiently large x,we have?g(p)(logp)e(?p1/2)?x3/4+?,where the implied constant depends on ?,?,? and the cusp form g. |
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