Localization of Divisors of Integers and of Some Arithmetic Functions

Abstract

We investigate several problems related to the multiplicative structure of integers. First, we determine the order of magnitude of the function H2(x, y, z), the number of positive integers n ≤ x having exactly two divisors in ( y, z], in the range of y10 ≤ z ≤ x1/3, and that of the function H3(x, y, z), the number of positive integers n ≤ x having exactly three divisors in ( y, z], in the range of y10 ≤ z ≤ x1/4. Next, we introduce a new function H1,1(x, y, z 1, z2), the number of positive integers n ≤ x having one divisor in (y, z 1] and one divisor in (z1, z 2] and study its order of magnitude. Finally, we investigate problems about localization of divisors of the Euler function &phis;(n) and the Carmichael function lambda(n). We say that a positive integer m is u-dense if whenever 1 ≤ y ≤ m, there is a divisor of m in the interval (y, uy]. We show that for > x integers n ≤ x, both &phis;(n) and lambda(n) are 2-dense.