View source for Dedekind sum

{{TEX|done}}{{MSC|11F20}}

Define $((x))$ for $x\in\mathbf R$ by $((m))=0$ if $m\in\mathbf Z$, and $((x+m))=x-1/2$ if $m\in\mathbf Z$, $0<x<1$. For integers $c$ and $d$, with $c>0$, the Dedekind sum $S(d,c)$ is the rational number defined by

$$S(d,c)=\sum_{x=1}^{c-1}\left(\left(\frac xc\right)\right)\left(\left(\frac{dx}{c}\right)\right).$$

R. Dedekind [[#References|[a1]]] showed that this quantity occurs in the transformation behaviour of the logarithm of the [[Dedekind eta-function|Dedekind eta-function]] under substitutions from the [[Modular group|modular group]]. This interpretation leads naturally to the reciprocity relation for Dedekind sums:

$$12S(d,c)+12S(c,d)=-3+\frac dc+\frac cd+\frac{1}{cd}$$

if $c>0$ and $d>0$ have [[Greatest common divisor|greatest common divisor]] $1$ (see also [[Quadratic reciprocity law|Quadratic reciprocity law]]). This relation resembles the reciprocity law for power-residue symbols. Several elementary proofs of this relation can be found in [[#References|[a2]]]. These proofs exhibit other interpretations of Dedekind sums, related to counting lattice points and Fourier theory (cf. [[Geometry of numbers|Geometry of numbers]]; [[Fourier transform|Fourier transform]]). There are many generalizations, see, e.g., [[#References|[a3]]], [[#References|[a4]]], [[#References|[a5]]], [[#References|[a6]]], [[#References|[a7]]].

====References====
<table>
<TR><TD valign="top">[a1]</TD> <TD valign="top">  R. Dedekind,   "Erläuterungen zu den fragmenten XXVIII"  H. Weber (ed.) , ''B. Riemann: Gesammelte mathematische Werke und wissenschaftlicher Nachlass'' , Dover, reprint  (1953)</TD></TR>
<TR><TD valign="top">[a2]</TD> <TD valign="top">  H. Rademacher,   E. Grosswald,   "Dedekind sums" , Math. Assoc. America  (1972)</TD></TR>
<TR><TD valign="top">[a3]</TD> <TD valign="top">  L.J. Goldstein,   "Dedekind sums for a Fuchsian group, I"  ''Nagoya Math. J.'' , '''50'''  (1973)  pp. 21–47</TD></TR>
<TR><TD valign="top">[a4]</TD> <TD valign="top">  L.J. Goldstein,   "Dedekind sums for a Fuchsian group, II"  ''Nagoya Math. J.'' , '''53'''  (1974)  pp. 171–187</TD></TR>
<TR><TD valign="top">[a5]</TD> <TD valign="top">  L.J. Goldstein,   "Errata for Dedekind sums for a Fuchsian group, I"  ''Nagoya Math. J.'' , '''53'''  (1974)  pp. 235–237</TD></TR>
<TR><TD valign="top">[a6]</TD> <TD valign="top">  U. Dieter,   "Cotangent sums, a further generalization of Dedekind sums"  ''J. Number Th.'' , '''18'''  (1984)  pp. 289–305</TD></TR>
<TR><TD valign="top">[a7]</TD> <TD valign="top">  R. Sczech,   "Dedekind symbols and power residue symbols"  ''Comp. Math.'' , '''59'''  (1986)  pp. 89–112</TD></TR>
</table>