Dedekind Sum

Given Relatively Prime Integers $p$ and $q$, the Dedekind sum is defined by

$$s(p,q)\equiv \sum_{i=1}^q \left(\!\!\left({i\over q}\right)\!\!\right)\left(\!\!\left({pi\over q}\right)\!\!\right),$$ (1)

where

$$((x))\equiv \cases{ x-\left\lfloor{x}\right\rfloor -{\textstyle{1\over 2}}& $x\not\in\Bbb{Z}$\cr 0 & $x\in\Bbb{Z}$.\cr}$$ (2)

Dedekind sums obey 2-term

$$s(p,q)+s(q,p)=-{1\over 4}+{1\over 12}\left({{p\over q}+{q\over p}+{1\over pq}}\right),$$ (3)

and 3-term

$$s(bc',a)+s(ca',b)+s(ab',c)=-{1\over 4}+{1\over 12}\left({{a\over bc}+{b\over ca}+{c\over ab}}\right)$$ (4)

reciprocity laws, where $a$, $b$, $c$ are pairwise Coprime and

$aa'\equiv 1\ \left({{\rm mod\ } {b}}\right)$	(5)
$bb'\equiv 1\ \left({{\rm mod\ } {c}}\right)$	(6)
$cc'\equiv 1\ \left({{\rm mod\ } {a}}\right)$	(7)

Let $p$, $q$, $u$, $v\in\Bbb{N}$ with $(p,q)=(u,v)=1$ (i.e., are pairwise Relatively Prime), then the Dedekind sums also satisfy

$$s(p,q)+s(u,v)=s(pu'-qv', pv+qu)-{\textstyle{1\over 4}}+{1\over 12}\left({{q\over vt}+{v\over tq}+{t\over qv}}\right),$$ (8)

where $t=pv+qu$, and $u'$, $v'$ are any Integers such that $uu'+vv'=1$ (Pommersheim 1993).

References

Pommersheim, J. ``Toric Varieties, Lattice Points, and Dedekind Sums.'' Math. Ann. 295, 1-24, 1993.