Topics related to the sum of unitary divisors of an integer

A divisor d of n is said to be a unitary divisor if d and n/d are relatively prime. Let σ*(n) be the sum of the unitary divisors of n, and let σ(n) be the sum of all the divisors of n. Some of the topics of classical number theory which involve σ(n) are investigated with the function σ replaced by σ*.

An integer n is said to be unitary perfect if σ*(n) = 2n ; some new results concerning such numbers are presented in Chapter II.

Two integers n and m are unitary amicable if they satisfy n + m = σ*(n) = σ*(m) Several theorems concerning unitary amicable numbers are proved in Chapter II, and an appendix lists 610 pairs of unitary amicable numbers.

Let D{X} be the asymptotic density of the set X of integers. It is known that the density function A(x) = D { n : σ(n)/n >x } ,, exists and is continuous for all values of the real variable x. Let ψ be Dedekind's function, ψ(n) = n p|n (1 + p ^-1) with the product over primes p which divide n. In Chapter III the existence and continuity of the density functions B(x) = D { n ψ(n)/n >x } and C(x) = D { n : σ(n)/n >x} is proved. In addition, upper and lower bounds are obtained for the functions B(x) and C(x) and, as a result, for A(x).