Dedekind sum

In mathematics, Dedekind sums are certain sums of products of a sawtooth function, and are given by a function D of three integer variables. Dedekind introduced them to express the functional equation of the Dedekind eta function. They have subsequently been much studied in number theory, and have occurred in some problems of topology. Dedekind sums have a large number functional equations; this article lists only a small fraction of these.

Dedekind sums were introduced by Richard Dedekind in a commentary on fragment XXVIII of Bernhard Riemann's collected papers.

Define the sawtooth function ${\displaystyle (\!(\,)\!):\mathbb {R} \rightarrow \mathbb {R} }$ as

We then let

be defined by

the terms on the right being the Dedekind sums. For the case a=1, one often writes

Note that D is symmetric in a and b, and hence

and that, by the oddness of (()),

By the periodicity of D in its first two arguments, the third argument being the length of the period for both,

If d is a positive integer, then

There is a proof for the last equality making use of

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