From wikipedia: "the average fraction by which the quotient n/k falls short of the next integer tends to γ". I was wondering if this theorem could be rephrased as showing a limiting distribution on [0,1] has mean 1-γ. I plotted histograms of the set of (X mod n)/n for all 1<=n<=X and fixed large X and they appear to converge to a decreasing/right skewed distribution on [0,1] with mean 1-γ.

Anybody know if this is known/could prove what it is?