On The Asymptotic Behavior Of Sums(?)

For any arbitrary real number a,we use {a} denote the fractional part of a.The study of the sum involving fractional part has always been concerned,because it has important connection with lots of questions in number theory.For example,according to Dirichlet's estimation of the partial sum of divisor function in 1849,we have(?) studing the order of the remainder term in the above sum is the famous Dirichlet's divisor problem.We can get many similar summations by the above formula,the common type is to add different weighted functions to the function {x/n} or change the power of it,namely the sum(?) The weighted function f(n)can be choosed arbitrarily,for the special case f(n)=na,Mercier used an elementary method to estimate the above sum and he got an asymptotic formula of it in 1985.However,Kolesnik used the method of Fourier series to get the upper bound of the sum(?) Also in 1985,Mercier and his collaborator Nowak considered the case that f(n)is nondecreasing.Based on the previous studies,this paper mainly considers the following question:let f(t)be an arbitrary real-valued positive nondecreasing function,let's start with the following two symbols(?) We prove that Sf,k(x)-Tf,k(x)=O(f(x)x131/416log26947/8320),Sf,k(x)-Tf,k(x)=Sf,1(x)-Tf,1(x)+0(f(x)x227/796+?),where k is a positive integer and e denotes a sufficiently small positive number.In addition,we also considered the estimation of the sum(?)as a further remark.