By accident I came across the following,

$$\lim_{n\to\infty}\frac{1}{n}\sum_{r=1}^n\frac{n\ (\mathrm{mod}\ r)}{r}=0.4227\ldots\approx 1-\gamma,$$

where the numerator is the remainder of $n$ divided by $r$. Is it known whether we have equality in the above expression or is it just a numerical coincidence? Has this been studied?

Edit: I'm sorry for all the (important) typos, everything should be fixed now.