Digit Variance and Dedekind Sums

Let p ⩾3 be a prime number, b ⩾2 a primitive root mod p and z an integer, 1⩽ z ⩽ p −1. The digit expansion of z / p with respect to the basis b has a period consisting of the first p −1 digits c 1 , …, c p −1 . We express the variance σ 2 of c 1 , …, c p −1 in terms of the Dedekind sum s ( p , b ) and investigate the behaviour of σ 2 for b fixed and p →∞. The reciprocity law for Dedekind sums is the most important tool in this investigation.