Difference between revisions of "Dedekind sum"
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− | Define | + | {{TEX|done}} |
+ | 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: | 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 | + | 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> | <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> |
Revision as of 19:02, 14 August 2014
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 |
Dedekind sum. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Dedekind_sum&oldid=32934