Note on a paper by G. J. Rieger

Let c is an element of IN. For d is an element of Z with god (d, c) = 1 let delta(d, c) be defined by d .delta(d, c) = 1 mod c, 1 less than or equal to delta(d, c) less than or equal to c. Let s, t is an element of IN with 1 less than or equal to s less than or equal to c, 1 less than or equal to t less than or equal to c. The main result is that for arbitrary fixed epsilon > 0, but uniformly over c, s and t, #{d is an element of IN: (c, d) = 1, 1 less than or equal to d less than or equal to s and 1 less than or equal to delta(d, c) less than or equal to t} = phi(c)/c(2)st + O-epsilon(c(1/2 + epsilon)). Equivalently, the points (d, delta(c, d)) are approximately uniformly distributed in [0, c] x [0, c]: The two-dimensional discrepancy of {(d, delta(c, d)): 1 less than or equal to d less than or equal to c and (c, d) = 1} in [0, c] x [0, c] is O-epsilon(c(epsilon-1/2)).