Peter wrote: Let $ a, b, n$ be positive integers with $ \gcd(a, b) = 1$. Prove that \[ \sum_{k}\left\{\frac {ak + b\}{n}\right\} = \frac {n - 1}{2}, \] where $ k$ runs through a complete system of residues modulo $ m$. I think it not true. For example $ a = n$ then $ \{\frac {ak + b}{n}\} = \{\frac {b}{n}\}$ If we chose $ b = 1$ then $ \sum_{k}\{\frac {ak + b}{n}\} = 1$ It is $ \gcd(a,n) = 1$ but it is quite easy. Because $ \{ak + b\}$ is a complete reside mod n when $ k$ run through an complete reside mod n.

And there is a typo too: the last $ m$ should be $ n$ in the statement. I will correct both for the next release. Thanks for reporting.