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

You could also ask this on Mathematics Stackexchange.
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When you say "average fraction" what is the distribution of such fractions?
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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.
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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.
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