On The Distribution Of The Number Of Square Factor And The Original Root

A positive integer n is called square-free number if it is not divisible by a perfect square except 1. For any integer n with (n,p)= 1, the smallest positive integer f such that nf= 1(mod p) is called the exponent of n modulo p. If the exponent of n modulo p is p - 1, then n is called a primitive root mod p. In this thesis we study the distribution of square-free numbers with a difference by using square sieve. What’s more, we study the distribution of consecutive square-free primitive roots modulo odd p. The results are as follows:1. Let S(n) be the characteristic function of the square-free numbers. In this thesis we mainly study the distribution of square-free numbers, improve the error term in the above, and investigate an asymptotic formula on frequency of square-free numbers with a given difference2. We study the distribution of consecutive square-free primitive roots mod-ulo p and give an asymptotic formula, by using properties of character sums. Let p be an odd prime, and let A(n) be the characteristic function of the square-free primitive roots modulo p. Then we have where φ(n) is Euler function, and ω(n) denotes the number of the distinct prime factors of n.