Resonance Between The Representation Function And Exponential Functions Over Arithmetic Progress

In analytic number theory,the problems concerning nonlinear exponential twisting arithmetic functions arise naturally in investigating equi-distribution theory,zero-distribution of L-functions and so on.Let an,be some arithmetic number-theoretic function,we usually consider the general nonlinear exponential sum of the form(?),0???R,0???1.Here,n?X means X?n?2X and e(z)?e2?iz.For an=A(n)and an??(n)(?is the Mobius function)the sums S(X,a)were studied by Iwaniec,Luo and Sarnak[2],and they showed these sums are intimately related to L-functions of GL2.If f is a holomorphic cusp form of even weight on the upper half plane,they also proved that a good upper bound of(X,?)implies a quasi Riemann hypothesis for L(s,f)[2].If ? is variable and an are the Fourier coefficients of automorphic forms.These sums are studied by Ren&Ye[3]and Sun&Wu[4].They proved that the resonance phenomenon occurs only when ?=1/2 and |?| is close to(?),k?Z+.Let r(n)denote the number of representations of a positive integer n as a sum of two squares,i.e.n=x12+x22,where x1,x2 are integers.Sun&Wu[4]also study the case that an=r(n),and obtain the resonance phenomenon.Yan[5]studied nonlinear exponential sum twisting the Fourier coefficients of Maass forms over the arithmetic progress and got an asymptotic formula(?),0???R,0???1.In this paper,we will consider the nonlinear exponential sum(?)where 0???R,0???1,e(x)=e2?ix and n?X means X<n?2X.Here X>1 is a large parameter.1?l?q are integers and(l,q)=1.We consider the case that q?? as X?? and obtain analogues of the result of Sun and Wu[4].The principal aim of this paper is to prove the following result.When the value of ???is small,we will get an upper bound estimation.Theorem 1.1.Let X>1,0<?<1 and 0???R.Let l,q?N and l?q?X1/2,the meaning of ?,D is in Lemma 2.1.If(?),then we have(?)When the value of ???is big,the result will be different.When ??1/2,we can still get an upper bound.Theorem 1.2.If(?)and??1/2.then we have(?)When?=1/2,we will discuss two situations according to the value of ???.We can get an upper bound of S(X)if ???is too big or small,and we will get an asymptotic formula if ???is appropriate.Theorem 1.3.If(?)and ?=1/2,then for ???<1/q or(?),we have where and ?c=1 or 0 according to if there exists an integer nc for c?q which satisfying or not In particular,we can see the resonance phenomenon occurs.Theorem 1.4.If(?)with 1?k?DX/4?,then we have To prove Theorem 1,we shall follow the steps in[3],[4]and[5].We will use a new Voronoi type summation formula generalized by Hu,Jiang&Lv[6].Thus we can get the Kloosterman sum and use the Weil's bound of it,so we can get the saving in the q-aspect and then obtain a similar main term of[5].In addition,we can see the resonance phenomenon also occurs for r(n)just like the Fourier coefficients of automorphic forms.