| In modern analytic number theory,to estimate the behaviors of Fourier co-efficients of various kinds of modular forms for GL(2)is an interesting research field.The famous Ramanujan-Petersson’s conjecture predicts that for any kind modular form g for GL(2),the order of Ag(n)does not exceed nε,where Ag(n)is the n-th Fourier coefficient of g,ε>0 is an arbitrary constant.When g is a holomorphic cusp form,this conjecture was proved by Deligne[5]via algebraic geometry methods.When g is an Eisenstein series,this conjecture was considered and proved by Eichler,Shimura,Ihara by means of representation and modular forms,(see[6],[33],[16],[17],etc.).When g is a general Maass cusp form,this conjecture has not been proved yet.But the Rankin-Selberg theory reveals that Ramanujan-Petersson’s conjecture is ture on average.In[8]Good proved that if g is a holomorphic cusp form or a Maass cusp form for S L(2,Z),thenComparing the above result with Ramanujan-Petersson’s conjectute,we see thatλg(n)oscillates with n.To study the oscillation behavior of λg(n),one usually considers the following sumwhere g is a cusp form for F0(D),X ≥ 1 is a large parameter,0 ≠α∈R and β>0.For D = 1(that is,g is a cusp form for S L(2,Z))in the sum(0.1),many authors obtained interesting results.Hafner[11]and Miller-Schmid[30]considered the case of linear exponential sum of(0.1)(i.e.β = 1)and proved that it is bounded by O(X1/2 + ε)uniformly for α∈R.As to the case of nonlinear exponential sums of(0.1),Ren-Ye[37]and Sun-Wu[34]proved that when 0<β<1 and β≠1/2,the sum(0.1)has an upper bound Oα(Xmax{β,1/2-β/4}+ε);when β = 1/2 and α is close towith q≤X/4,the sum(0.1)has a main term of size Iλg(q)|q-1/4X3/4.Their result was motivated by the work of Iwaniec,Luo and Samak in[20],where the asymptotic formula of(0.1)was considered for the first time when β=1/2,with q∈N.For D>1,Harcos pointed out that the sum(0.1)is bounded by 0((DX)1/2+E)uniformly for or ∈ R in[13].However,no asymptotic result of(0.1)has been obtained so far.In this paper,we will consider a more general exponential sum twisted by Fourier coefficients of cusp forms.Let g be a primitive newform of level D and nebentypus χD(n).Precisely,we study the sum of the form belowwhere 0≠α∈R,0＜β＜1,l and N are coprime integers,(D,N)= 1 or D is square-free.In paticular,when N=1,the sum(0.2)becomes(0.1).According to g is a holomorphic cusp form or a Maass cusp form,we prove the following results respectively.Theorem 1 Let N2D ≤ X1-ε,g be a Maass cusp form of weight 0,Laplace eigen-value v2 + 1/4,0>0 be the smallest real number such that λg(n)《nθ for any n≠0.(1)Assume D1/2N|α|βXβ</2,then we have(2)Assume D1/2N|α|βXβ≥/2.where cα is a constant,c0 =1+i,D2 ＝D/(c,D).Here δc is one or zero according to whether or not |n0-|α|2c2D2/4| ≤ X-ε with n0 = n0(c)is the positive integer nearest to(|α|c)2D2/4,ηg(D2),λgD2(q)are as defined in(2.10),is as defined in(3.8)and ε(α,n0,c,D2,X)is as defined in(3.28).In particular,if with integer q satisfying 1≤q≤ X/(4DN3),then we haveTheorem 2 In Theorem 1,if we change g into a holomorphic cusp form of weight k and change the condition N2D ≤ X1-ε into N2D ≤ X,then the results in(0.3),(0.4),(0.5),(0.6),(0.7)still hold with constant ca replaced by a constantand Ov,ε replaced byand Oκ,ε respectively and θ replaced by s.When D = N= 1,Theorem 1 covers the result of Ren and Ye in[37];Theorem 2 covers the result of Sun and Wu in[34].When X is sufficiently large,β = 1/2,a is close to,the sum SD(N,α,β,X)has an asymptotic formula.This is the first asymptotic result of the exponential sums twisted by Fourier coefficients of cusp forms for Γ0(D)(D>1).We are also interested in the estimate of exponential sums twisted by Fourier coefficients of Eisenstein series for Γ0(D).For Eisenstein series of level D = 1 and weight zero E(z,s),Hardy[14]and Uchiyama[38]considered exponential sums twisted by Fourier coefficients of E’(z,s)|s=1/2 in natural numbers and arith-metic progressions,respectively.They proved that when β＝1/2 and a equals to some special values,the exponential sums in natural numbers and arithmetic progressions both have asymptotic formulas.In this paper,we consider the case of D>1.Let g =E0χD’,χD" be a non-holomorphic Eisenstein series of Laplace eigenvalue 1/4,weight 0,level D = D’D" ≥ 2 and nebentypus χD=XD’XD" With(D’,D")= 1 and χD is a primitive character.Then its n-th Fourier coefficient isWe will estimate the exponential sum belowwhere 0≠α∈R,0＜β＜1,l and N are coprime integers and N2D ≤ X1-ε.Theorem 3 In Theorem 1,if we change SD(N,α,β,X)into UχD’χD"(N,α,β,X),then the results(0.3),(0.4),(0.5)(0.6)and(0.7)still hold with θ replaced byand Ov,ε replaced byand Oε respectively,the definition of ηg(D2)and gD2 re-placed by(2.12)and(2.13)respectively and an extra termadded in all upper bounds of(0.3),(0.4),(0.5)and error terms of(0.6)and(0.7).When X is sufficiently large,β= 1/2,α is close to,the sum(0.8)has an asymptotic formula.This is the first asymptotic result of the exponential sum twisted by Fourier coefficients of Eisentein series for Γ0(D)(D>1).The main tools to prove the above theorems are Voronoi summation formulas and stationary phase arguments.Before applying Voronoi summation formulas,we need to change the clear-cut sum into a smooth sum,that is,introduce a smooth weight function supported on interval[X,2X].Thus the main part of the proof of Theorem 1 and Theorem 2 is to estimate the exponential sum belowwhere φ(x)is a smooth function supported on inteval[1,2],α,β,g,l,N are the same as stated in the sum(0.2).Similar to Theorem 1 and Theorem 2,for β＝1/2 and a is close to,the sum(0.9)has an asymptotic formula;for other 0<<1 and α∈R,the sum(0.9)has nontrivial upper bounds.Specifically,when,the sum(0.9)has an upper boundBut for some special α and β,we can improve the result(0.10)and obtained the following results.Theorem 4 Let 1≤D≤X1-ε,g be a holomorphic or a Maass primitive newform for Γ0(D),integer s≥ 2 and φ(x)is a smooth function supported on interval[1,2]such thatfor any r ≥ 0.When |α|2s-1 is rational,the fractional part ofis α/q such that(α,q)＝1,α,q∈Z.If q satisfies,we haveTheorem 5 Let N2D ≤ X1-ε,g be a holomorphic or Maass cusp form for S L(2,Z)and φ(x)be as stated in Theorem 4.Then for any α∈R,we haveComparing the results in Theorems 4 and 5 with that Theorems 1 and 2,we see that the results in Theorems 4 and 5 improve the corresponding results in Theorems 1 and 2.The main idea to prove Theorems 4 and 5 is change the order of the exponential sum twisted by λg(n)form β to β/(2β-1)by stationary phase argument.We point out that for the clear-cut exponential sum(i.e.the sum(0.2))we cannot get similar results as stated in Theorems 4 and 5.The reason is that a larger term will be produced when we try to remove the weight. |
PDF Full Text Request