The fact that the average value of \(N_n\) is approximately \((12\log 2/\pi^2)\log n\) was previously known from work of Heilbronn and Dixon in 1969--1970, using ideas dating back to Gauss, who discov...The fact that the average value of \(N_n\) is approximately \((12\log 2/\pi^2)\log n\) was previously known from work of Heilbronn and Dixon in 1969--1970, using ideas dating back to Gauss, who discovered the probability distribution now called the ``Gauss measure''. This is the probability distribution on \((0,1)\) with density \(\frac{1}{\log 2(1+x)}\), which Gauss found (but did not prove!) describes the limiting distribution of the ratio of a pair of independent \(U(0,1)\) random variables …