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 zΓp 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 _ 2 of c 1 , ..., c p&1 in terms of the Dedekind sum s( p, b) and investigate the behaviour of _ 2 for b fixed and p Γ . The reciprocity law for Dedekind sums is the most important tool in this investigation.
1997 Academic Press
1. MAIN RESULTS
The classical Dedekind sums s(b, n) go back to the 19th century and can be defined in an elementary way (cf. formula (4) below). Originally, however, they did not occur in an elementary context but in connection with deep results of complex analysis. Methods of complex analysis also supplied the first proof of the most important property of Dedekind sums, the so-called reciprocity law (cf. [5, Chap. 6]). In the 20th century a number of elementary proofs of this law were given. Moreover, it turned out that Dedekind sums have a meaning in an elementary mathematical context, too (cf. [2, p. 162ff; 5, p. 35ff]). In this article we exhibit another meaning, which is perhaps the most elementary considered so far.
Let b and n be natural numbers 2, and z an integer with 1 z n&1, (z, n)=1. We consider the digit expansion of the rational number zΓn with respect to the basis b, i.e.,
where c j is one of 0, 1, ..., b&1, and c j is different from b&1 for infinitely many indices j. In this way the sequence of digits c j , j=1, 2, 3, ..., is uniquely determined. Of course, it is a periodic sequence, and it has no preperiod if (b, n)=1. In the sequel we always make this assumption. Let