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, Erdős, Pomerance, and Straus, according to which the set of integers n such that is an integer is of density 1/2. Research supported in part by a grant from NSERC. Research supported by the Applied Number Theory Research Group of the Hungarian Academy of Science and by a grant from OTKA. Published in Lietuvos Matematikos Rinkinys, Vol. 46, No. 3, pp. 315–331, July–September, 2006.