Estimates Of Exponential Sums Twisted By Fourier Coefficients Of Modular Forms For ?₀?D? In Arithmetic Progressions

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 n?,where Ag(n)is the n-th Fourier coefficient of g,?>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 that?g(n)oscillates with n.To study the oscillation behavior of ?g(n),one usually considers the following sumwhere g is a cusp form for F0(D),X ? 1 is a large parameter,0 ???R and ?>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.? = 1)and proved that it is bounded by O(X1/2 + ?)uniformly for ??R.As to the case of nonlinear exponential sums of(0.1),Ren-Ye[37]and Sun-Wu[34]proved that when 0<?<1 and ??1/2,the sum(0.1)has an upper bound O?(Xmax{?,1/2-?/4}+?);when ? = 1/2 and ? is close towith q?X/4,the sum(0.1)has a main term of size I?g(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 ?=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 ?D(n).Precisely,we study the sum of the form belowwhere 0???R,0???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-?,g be a Maass cusp form of weight 0,Laplace eigen-value v2 + 1/4,0>0 be the smallest real number such that ?g(n)?n? for any n?0.(1)Assume D1/2N|?|?X?</2,then we have(2)Assume D1/2N|?|?X??/2.where c? is a constant,c0 =1+i,D2 ?D/(c,D).Here ?c is one or zero according to whether or not |n0-|?|2c2D2/4| ? X-? with n0 = n0(c)is the positive integer nearest to(|?|c)2D2/4,?g(D2),?gD2(q)are as defined in(2.10),is as defined in(3.8)and ?(?,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-? 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,? replaced byand O?,? respectively and ? 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,? = 1/2,a is close to,the sum SD(N,?,?,X)has an asymptotic formula.This is the first asymptotic result of the exponential sums twisted by Fourier coefficients of cusp forms for ?0(D)(D>1).We are also interested in the estimate of exponential sums twisted by Fourier coefficients of Eisenstein series for ?0(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 ??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 =E0?D',?D" be a non-holomorphic Eisenstein series of Laplace eigenvalue 1/4,weight 0,level D = D'D" ? 2 and nebentypus ?D=XD'XD" With(D',D")= 1 and ?D is a primitive character.Then its n-th Fourier coefficient isWe will estimate the exponential sum belowwhere 0???R,0???1,l and N are coprime integers and N2D ? X1-?.Theorem 3 In Theorem 1,if we change SD(N,?,?,X)into U?D'?D"(N,?,?,X),then the results(0.3),(0.4),(0.5)(0.6)and(0.7)still hold with ? replaced byand Ov,? replaced byand O? respectively,the definition of ?g(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,?= 1/2,? 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 ?0(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 ?(x)is a smooth function supported on inteval[1,2],?,?,g,l,N are the same as stated in the sum(0.2).Similar to Theorem 1 and Theorem 2,for ??1/2 and a is close to,the sum(0.9)has an asymptotic formula;for other 0<<1 and ??R,the sum(0.9)has nontrivial upper bounds.Specifically,when,the sum(0.9)has an upper boundBut for some special ? and ?,we can improve the result(0.10)and obtained the following results.Theorem 4 Let 1?D?X1-?,g be a holomorphic or a Maass primitive newform for ?0(D),integer s? 2 and ?(x)is a smooth function supported on interval[1,2]such thatfor any r ? 0.When |?|2s-1 is rational,the fractional part ofis ?/q such that(?,q)?1,?,q?Z.If q satisfies,we haveTheorem 5 Let N2D ? X1-?,g be a holomorphic or Maass cusp form for S L(2,Z)and ?(x)be as stated in Theorem 4.Then for any ??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 ?g(n)form ? to ?/(2?-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.