设 p 为一个正整数, 如果对任意整除 n 的素数 p, 一定有 p 的平方不整除 n, 那么就称 n 为一个无平方因子数. 我们将考虑无平方因子数在算术序列中的分布问题. 具体来讲, 设 p 为正整数, 对 gcd(a,p)=1, 记 S(x;a,q) 为算术序列 kq+a, k=1,2,3··· 中不超过正数 x 的无平方因子数的个数. 我们将研究 S(x;a,q) 的渐近性质, 期望改进Hooley的一个结果.

Let n be an positive integer. If for any prime number p dividing n, its square can not divide n. Then we say that n is a squarefree number. We will consider the distribution of squarefree numbers in the arithmetic progression. Specifically, for an positive integer q and gcd(a,q)=1, let S(x;a,q) be the number of squarefree numbers which are in the the arithmetic progression kq+a, k=1,2,3··· and don't exceed x. We will study the property of S(x;a,q) and hope to improve an result of Hooley.