Number Of Special Features, And Estimate The Class Set

The problem of the Dirichlet character sums is one of the center content in analytic number theory research. Many scholars have done a lot of researches about this issue and got many important results. Polya, Vinogradov and Burgess studied the estimate of character sums on the continuous integer numbers. On the basis of their results, many scholars study the Dirichlet character sums over the special numbers, such as the free square numbers, smooth numbers, facto-rials numbers, binomial coefficients and other combinatorial numbers. In this paper, we study the arithmetic nature of a number set related to the smooth number, and certain character sums by some elementary and analytic methods or techniques. The main content of this paper is as follows:1. S(n,y)={x|1≤x≤n,p(x)≤y} denote the y-smooth numbers, where P(x) is the largest divisor of x (with the usual convention that P(1)=1). We defineAy,={n:n=m+r,m≤n<(m+1),r∈(n, y),n∈N}. The main purpose of this paper is to give the asymptotic behavior of the number and the average of Ay.2. Let q≥2be an positive integer,X be a non-principle character mod q, A=A(q)<q, B=B(q)<q and H=H(q)<q. In this paper, we will use the estimates for Kloosterman’s sums and character sums, and properties of trigonometric sums to give a sharp asymptotic estimate for the certain character sums of the formTk(x,A,BtH;q)=akX(a) a∈h(A,B,H) 8/*and wher76亢(A,B,H)={a∈Z(a,q):1,1≤a≤A,1≤b≤B,ab≡1(mod g),|a-b|≤H}.