Journal of Integer Sequences, Vol. 17 (2014), Article 14.7.7

On the Asymptotic Behavior of Dedekind Sums

Kurt Girstmair
Institut für Mathematik
Universität Innsbruck
Technikerstr. 13/7
A-6020 Innsbruck


Let z be a real quadratic irrational. We compare the asymptotic behavior of Dedekind sums S(pk, qk) belonging to convergents pk/qk of the regular continued fraction expansion of z with that of Dedekind sums S(sj, tj) belonging to convergents sj/tj of the negative regular continued fraction expansion of z. Whereas the three main cases of this behavior are closely related, a more detailed study of the most interesting case (in which the Dedekind sums remain bounded) exhibits some marked differences, since the cluster points depend on the respective periods of these expansions. We show in which cases cluster points of S(sj, tj) can coincide with cluster points of S(pk, qk). An important tool for our purpose is a criterion that says which convergents sj/tj of z are convergents pk/qk.

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Received March 14 2014; revised version received July 3 2014. Published in Journal of Integer Sequences, July 11 2014.

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