On Abundant Numbers

We say that the positive integer n is abundant, when

σ(n) = Σd≥2n.

The probability that a positive integer is abundant is less than 1/2. More precisely if A(x) is the number of solutions of

σ(n)≥2n (l≤n≤x) then

0 < A(x)/x <1 for all x > x₀.