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?
Interesting question, I'd really like to see your plot.

You could also ask this on Mathematics Stackexchange.
When you say "average fraction" what is the distribution of such fractions?
Most likely, this can be turned into the integral of floor(1/x) - 1/x from 0 to 1, which does indeed take the value 1-γ.
If there is a source on Wikipedia you should look it up.

> "the average fraction (...) tends to γ".

this reads really weird, since γ is not a fraction. But most likely the precise statement is something like:

lim\_{N->infty} 1/N\^2  sum\_{q<N}  sum\_{a mod q} ||a/q|| = γ.

where this ||.|| absolute value is the distance to the nearest integer.