Geodesic
On the asymptotic behavior of Dedekind sums
Journal of integer sequences, Tome 17 (2014) no. 7.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

Summary: Let $z$ be a real quadratic irrational. We compare the asymptotic behavior of Dedekind sums $S(p_{k}, q_{k})$ belonging to convergents $p_{k}/q_{k}$ of the $regular$ continued fraction expansion of $z$ with that of Dedekind sums $S(s_{j}, t_{j})$ belonging to convergents $s_{j}/t_{j}$ 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 $S(s_{j}, t_{j})$ can coincide with cluster points of $S(p_{k}, q_{k})$. An important tool for our purpose is a criterion that says which convergents $s_{j}/t_{j}$ of $z$ are convergents $p_{k}/q_{k}$.
Keywords: asymptotic behavior of Dedekind sums, continued fraction expansion of a quadratic irrational 
@article{JIS_2014__17_7_a4,
     author = {Girstmair, Kurt},
     title = {On the asymptotic behavior of {Dedekind} sums},
     journal = {Journal of integer sequences},
     publisher = {mathdoc},
     volume = {17},
     number = {7},
     year = {2014},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JIS_2014__17_7_a4/}
}
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Girstmair, Kurt. On the asymptotic behavior of Dedekind sums. Journal of integer sequences, Tome 17 (2014) no. 7. http://geodesic.mathdoc.fr/item/JIS_2014__17_7_a4/