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Define <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110070/d1100701.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110070/d1100702.png" /> by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110070/d1100703.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110070/d1100704.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110070/d1100705.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110070/d1100706.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110070/d1100707.png" />. For integers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110070/d1100708.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110070/d1100709.png" />, with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110070/d11007010.png" />, the Dedekind sum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110070/d11007011.png" /> is the rational number defined by
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{{TEX|done}}{{MSC|11F20}}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110070/d11007012.png" /></td> </tr></table>
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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
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$$S(d,c)=\sum_{x=1}^{c-1}\left(\left(\frac xc\right)\right)\left(\left(\frac{dx}{c}\right)\right).$$
  
 
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:
 
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:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110070/d11007013.png" /></td> </tr></table>
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$$12S(d,c)+12S(c,d)=-3+\frac dc+\frac cd+\frac{1}{cd}$$
  
if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110070/d11007014.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110070/d11007015.png" /> have [[Greatest common divisor|greatest common divisor]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110070/d11007016.png" /> (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]]].
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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====
 
====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>
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<table>
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<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>
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<TR><TD valign="top">[a2]</TD> <TD valign="top">  H. Rademacher,  E. Grosswald,  "Dedekind sums" , Math. Assoc. America  (1972)</TD></TR>
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<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>
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<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>
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<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>
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<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>
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<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>
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</table>

Latest revision as of 21:39, 23 December 2015

2020 Mathematics Subject Classification: Primary: 11F20 [MSN][ZBL]

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 [a1] showed that this quantity occurs in the transformation behaviour of the logarithm of the Dedekind eta-function under substitutions from the 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 $1$ (see also Quadratic reciprocity law). This relation resembles the reciprocity law for power-residue symbols. Several elementary proofs of this relation can be found in [a2]. These proofs exhibit other interpretations of Dedekind sums, related to counting lattice points and Fourier theory (cf. Geometry of numbers; Fourier transform). There are many generalizations, see, e.g., [a3], [a4], [a5], [a6], [a7].

References

[a1] R. Dedekind, "Erläuterungen zu den fragmenten XXVIII" H. Weber (ed.) , B. Riemann: Gesammelte mathematische Werke und wissenschaftlicher Nachlass , Dover, reprint (1953)
[a2] H. Rademacher, E. Grosswald, "Dedekind sums" , Math. Assoc. America (1972)
[a3] L.J. Goldstein, "Dedekind sums for a Fuchsian group, I" Nagoya Math. J. , 50 (1973) pp. 21–47
[a4] L.J. Goldstein, "Dedekind sums for a Fuchsian group, II" Nagoya Math. J. , 53 (1974) pp. 171–187
[a5] L.J. Goldstein, "Errata for Dedekind sums for a Fuchsian group, I" Nagoya Math. J. , 53 (1974) pp. 235–237
[a6] U. Dieter, "Cotangent sums, a further generalization of Dedekind sums" J. Number Th. , 18 (1984) pp. 289–305
[a7] R. Sczech, "Dedekind symbols and power residue symbols" Comp. Math. , 59 (1986) pp. 89–112
How to Cite This Entry:
Dedekind sum. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Dedekind_sum&oldid=13005
This article was adapted from an original article by R.W. Bruggeman (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article