Equidistribution Mod Q of Abundant and Deficient Numbers

The ancient Greeks called the natural number m deficient, perfect, or abundant according to whether σ(m) 2m. In 1933, Davenport showed that all three of these sets make up a well-defined proportion of the positive integers. More precisely, if we letm D(u; x) := m ≤ x : ≤ u , and put D(u; x) := #D(u; x), σ(m) then Davenport’s theorem asserts that limx→∞ x 1 D(u; x) exists for every u. More- over, D(u) is a continuous function of u, with D(0) = 0 and D(1) = 1. In this note, we study the distribution of D(u; x) in arithmetic progressions. A simple to state consequence of our main result is the following: Fix u ∈ (0, 1]. Then the elements of D(u; x) approach equidistribution modulo prime numbers q whenever q, x, and x all tend to infinity. q log log log x