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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),
\end{displaymath} (1)

((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}
\end{displaymath} (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),
\end{displaymath} (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)
\end{displaymath} (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),
\end{displaymath} (8)

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


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

© 1996-9 Eric W. Weisstein