Dedekind sum

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