Sharp asymptotic order of amicable numbers

Determine whether the counting function A(x) for amicable numbers satisfies the asymptotic formula A(x)=x/L(x)^{2+o(1)} as x\to\infty, where L(x)=\exp\!\bigl(\log x\,\log_3 x/\log_2 x\bigr).

Background

The main theorem establishes the upper bound A(x)\le xL(x)^{-1/2+o(1)}. The introduction argues heuristically that the exponent 1/2 is not expected to be optimal and that the exponent 2 may describe the true order of magnitude.

The proposed asymptotic is motivated by estimates for the second moment of the number of solutions to \sigma(m)=n and by the heuristic that the additional relation n+n'=m holds for roughly a proportion x^{-1} of relevant pairs. The paper does not prove this sharper estimate and presents it as plausible rather than established.