How quadratic are the natural numbers?

For natural numbers n we inspect all factorizations n = ab of n with a ≦ b in double-struck N sign and denote by n = a_nb_n the most quadratic one, i.e. such that b_n - a_n is minimal. Then the quotient κ(n) := a_n/b_n is a measure for the quadraticity of n. The best general estimate for κ(n) is of course very poor: 1/n ≦ κ(n) ≦ 1. But a Theorem of Hall and Tenenbaum [1, p. 29], implies (log n)^-δ-ε ≦ κ(n) ≦ (log n)^-δ on average, with δ = 1 - (1 + log₂ 2)/log2 = 0,08607 ... and for every ε > 0. Hence the natural numbers are fairly quadratic. κ(n) characterizes a specific optimal factorization of n. A quadraticity measure, which is more global with respect to the prime factorization of n, is κ^*(n) := Σ1 ≦ n ≦ b ,ab = n a/b. We show κ^*(n) ∼ 1/2 on average, and κ^*(n) = Ω(2^{1/2(1-ε)log n/log2n}) for every ε > 0.