We study the behaviour of the classical Dedekind sums s ( m / n ) for convergents m / n of a given quadratic irrational α . It turns out that two cases may occur: Either the sequence s ( m / n ) remains bounded with finitely many cluster points, or s ( m / n ) tends to +∞ or −∞ like + log n or − log n , respectively. By means of the Barkan–Hickerson–Knuth formula we obtain a precise description of what happens in all cases.
We study the asymptotic behaviour of the classical Dedekind sums s(m/n) for convergents m/n of e, e 2 , and (e+1)/(e−1), where e = 2.71828 . . . is Euler’s number. Our main tool is the Barkan-Hickerson-Knuth formula, which yields a precise description of what happens in all cases.
This paper has been withdrawn by the author due to a crucial mistakes.