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(?n?)where 0? ? ? R and 0<?<1.An asymptotic formula is established for the nonlinear exponential sum when ?=1/2 and |?| is close to(?),? ?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)<<?1/4+?U1/4(1<<U<<x3/5),and?(x+U)-?(x)<<?x2/9+?U1/3(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(?,?)is an exponent pair for which ??0,then for X?-?? V3+2?-2?U-2,we have R<<?X?(XV-5U2 + X?+?/?U2?+2/?V-3+4?+2?/?).In 1916,Hardy[2]studied the sum and showed that,if t ? 4?q1/2 for any positive integer q,then S(X,t)=o(X?)and,if t= 4?q1/2 for some integer q,then S(X,t)= 2(1+i)d(q)/q1/4X1/4+0(X?)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 |?|?X?<(?),then and when |?|?X??(?),thenwhere K?,q=1 or 0 according to if there exists a positive integer q satisfying||?|-2(?)?X-1/2,1?|?|<(?),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 ??(?)for any integer k,then U(?,X)=O?(log X),and if ?=(?)for some integer k,then provided that ??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 ?=1/2 and |?|is close(?),?Z+,and obtain a new result when ??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<?<1 and 0 ? ? ? R.Suppose also l,q ?N and l?q?X1/2.For|?|?X?<(?),one hasTheorem 2.Suppose X>1,0<?<1 and 0 ? ? ? R.Suppose also l,q ?N and l ? q? X1/2.For |?|?X??(?)and ? ?1/2,one hasTheorem 3.Suppose X>1,0<?<1 and 0 ? ? ? R.Suppose also l,q ? N and l ? q? X1/2.For |?|?X??(?)and ?=1/2,if |?|<1/q or |?|?(?)one has if 1/q ?<?|?(?)one has where and ?k=1 or 0 according to whether there is a positive integer nk for k|q satisfying or not.