Resonance Between Exponential Functions And Divisor Function Over Arithmetic Progressions

We study the resonance phenomenon between divisor function and exponen-tial functions of the form e(αnβ)where 0≠ α ∈ R and 0<β<1.An asymptotic formula is established for the nonlinear exponential sum when β=1/2 and |α| is close to(?),κ ∈Z+.Let d(n)denote the number of divisors of n.The properties of divisor function d(n)attract many researchers’ interest.Aleksandar Ivic and Wenguang Zhai[1]studied the Dirichlet divisor problem in short intervals,and showed that△(x+U)-△(x)<<ε1/4+εU1/4(1<<U<<x3/5),and△(x+U)-△(x)<<εx2/9+εU1/3(1<<V<<x2/3),where U = o(x)and Moreover,suppose that|△(xr+U)-△(xr)|≥V>>U1/2(>>1)(r=1,...,R-1),where X/2 ≤ x1<...<xR ≤ X,|xr-xs| ≥ V,r≠s.If(κ,λ)is an exponent pair for which κ≠0,then for Xλ-κ≤ V3+2λ-2κU-2,we have R<<εXε(XV-5U2 + Xκ+λ/κU2κ+2/κV-3+4κ+2λ/κ).In 1916,Hardy[2]studied the sum and showed that,if t ≠ 4πq1/2 for any positive integer q,then S(X,t)=o(Xε)and,if t= 4πq1/2 for some integer q,then S(X,t)= 2(1+i)d(q)/q1/4X1/4+0(Xε)as X →∞.The above result can be seen as the resonance phenomenon between divisor function and exponential functions.Recently,Sun and Wu[3]considered a similar problem and proved when |α|βXβ<(?),then and when |α|βXβ≥(?),thenwhere Kα,q=1 or 0 according to if there exists a positive integer q satisfying||α|-2(?)≤X-1/2,1≤|α|<(?),and In 1973,Saburo Uchiyama[4]considered the sum where q and l are integers with q ≥ 1,0 ≤ l<q,and showed that,if α≠(?)for any integer k,then U(α,X)=Oα(log X),and if α=(?)for some integer k,then provided that α≥4q3,whereand S(m,n;q)denotes the Kloosterman sum.Motivated by the above results,we study the resonance phenomenon between divisor function and exponential functions over arithmetic progressions.To do this,using a different method we establish an asymptotic formula when β=1/2 and |α|is close(?),∈Z+,and obtain a new result when β≠1/2.Our main instru-ments are Voronoi formula,the estimation of exponential sums and the weighted stationary phase.Due to the orthogonality of the additive characters,we can obtain the Kloosterman sum after applying the Voronoi summation formula,which let us can use Weil’s bound to get the saving in the q-aspect.Our result is the following theorem.Theorem 1.Suppose X>1,0<β<1 and 0 ≠ α ∈ R.Suppose also l,q ∈N and l≤q≤X1/2.For|α|βXβ<(?),one hasTheorem 2.Suppose X>1,0<β<1 and 0 ≠ α ∈ R.Suppose also l,q ∈N and l ≤ q≤ X1/2.For |α|βXβ≥(?)and β ≠1/2,one hasTheorem 3.Suppose X>1,0<β<1 and 0 ≠ α ∈ R.Suppose also l,q ∈ N and l ≤ q≤ X1/2.For |α|βXβ≥(?)and β=1/2,if |α|<1/q or |α|≥(?)one has if 1/q ≤<α|≤(?)one has where and δk=1 or 0 according to whether there is a positive integer nk for k|q satisfying or not.