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 obey a large number of relationships on themselves; this article lists only a tiny fraction of these. dedekind sums were introduced by richard dedekind in a commentary on fragment xxviii of bernhard riemann's collected papers. == definition == define the sawtooth function \left( \left( \right) \right):\mathbb{r} \rightarrow \mathbb{r} as :((x))=\begin{cases} x-\lfloor x\rfloor - 1/2, &\mbox{if }x\in\mathbb{r}\setminus\mathbb{z};\\ 0,&\mbox{if }x\in\mathbb{z}. \end{cases} we then let :d :z3 &rarr; r be defined by :d(a,b;c)=\sum_{n \bmod c} \left( \bigg( \frac{an}{c} \bigg) \right) \left( \left( \frac{bn}{c} \right) \right), the terms on the right being the dedekind sums. for the case a=1, one often writes :s(b,c) = d(1,b;c). ==simple formulae== note that d is symmetric in a and b, and hence :d(a,b;c)=d(b,a;c), and that, by the oddness of (()), :d(&minus;a,b;c) = &minus;d(a,b;c), :d(a,b;&minus;c) = d(a,b;c). by the periodicity of d in its first two arguments, the third argument being the length of the period for both, :d(a,b;c)=d(a+kc,b+lc;c), for all integers k,l. if d is a positive integer, then :d(ad,bd;cd) = dd(a,b;c), :d(ad,bd;c) = d(a,b;c), if (d,c) = 1, :d(ad,b;cd) = d(a,b;c), if (d,b) = 1. there is a proof for the last equality making use of :\sum_{n \bmod c} \left( \left( \frac{n+x}{c} \right) \right)=\left(\left( x\right)\right),\qquad\forall x\in\mathbb{r}. furthermore, az = 1 (mod c) implies d(a,b;c) = d(1,bz;c). ==alternative forms== if b and c are coprime, we may write s(b,c) as :s(b,c)=\frac{-1}{c} \sum_\omega \frac{1} { (1-\omega^b) (1-\omega ) } +\frac{1}{4} - \frac{1}{4c}, where the sum extends over the c-th roots of unity other than 1, i.e. over all \omega such that \omega^c=1 and \omega\not=1. if b, c &gt; 0 are coprime, then :s(b,c)=\frac{1}{4c}\sum_{n=1}^{c-1} \cot \left( \frac{\pi n}{c} \right) \cot \left( \frac{\pi nb}{c} \right). ==reciprocity law== if b and c are coprime positive integers then :s(b,c)+s(c,b) =\frac{1}{12}\left(\frac{b}{c}+\frac{1}{bc}+\frac{c}{b}\right)-\frac{1}{4}. rewriting this as :12bc \left( s(b,c) + s(c,b) \right) = b^2 + c^2 -3bc + 1, it follows that the number 6c&nbsp;s(b,c) is an integer. if k
Modular_forms Number_theory