On the local distribution of certain arithmetic functions

Let d(n), σ 1(n), and φ(n) stand for the number of positive divisors of n, the sum of the positive divisors of n, and Euler's function, respectively. For each ν, Z, we obtain asymptotic formulas for the number of integers n ≥ x for which e n = 2 v r for some odd integer m as well as for the number of integers n ≥ x for which e n = 2 v r for some odd rational number r. Our method also applies when φ(n) is replaced by σ 1(n), thus, improving upon an earlier result of Bateman, Erdos, Pomerance, and Straus, according to which the set of integers n such that σ1(n)/d(n)2 is an integer is of density 1/2. © 2006 Springer Science+Business Media, Inc.