Probability distribution for the remainder of a fixed integer In the "Notes" section of Modern Computer Algebra by Joachim Von Zur Gathen, there is a quick throwaway remark that says:

Dirichlet also proves the fact, surprising at first sight, that for fixed $a$ in a division the remainder $r = a \operatorname{rem} b$, with $0 \leq r < b$, is more likely to be smaller than $b/2$ than larger: If $p_a$ denotes the probability for the former, where $1 \leq b \leq a$ is chosen uniformly at random, then $p_a$ is asymptotically $2 - \ln{4} \approx 61.37\%$.

The note ends there and nothing is said about it again. This fact does surprise me, and I've tried to look it up, but all my searches for "Dirichlet" and "probability" together end up being dominated by talks of Dirichlet stochastic processes (which, I assume, is unrelated).
Does anybody have a reference or proof for this result?

The Prime Numbers and Their Distribution By Gérald Tenenbaum, Michel Mendès France
Parts of this book, including discussion of the result, are available on Google Books.
