Inmathematics,Dedekind sums are certain sums of products of asawtooth function, and are given by afunctionD of threeinteger variables.Dedekind introduced them to express thefunctional equation of theDedekind eta function. They have subsequently been much studied innumber theory, and have occurred in some problems oftopology. Dedekind sums have a large number of functional equations; this article lists only a small fraction of these.

Dedekind sums were introduced byRichard Dedekind in a commentary on fragment XXVIII ofBernhard Riemann's collected papers.

Define thesawtooth function${\displaystyle (()):\mathbb{R}\to \mathbb{R}}$ as

We then let

${\displaystyle D:{\mathbb{Z}}^2\times (\mathbb{Z}-\{0\})\to \mathbb{R}}$

be defined by

${\displaystyle D(a,b;c)=\sum _{n=1}^{c-1}\left(\left(\frac{an}{c}\right)\right)\left(\left(\frac{bn}{c}\right)\right),}$

the terms on the right being theDedekind sums. For the casea = 1, one often writes

s(b,c) =D(1,b;c).

Note thatD is symmetric ina andb, and hence

${\displaystyle D(a,b;c)=D(b,a;c),}$

and that, by the oddness of (( )),

D(−a,b;c) = −D(a,b;c),

D(a,b; −c) =D(a,b;c).

By the periodicity ofD in its first two arguments, the third argument being the length of the period for both,

D(a,b;c) =D(a+kc,b+lc;c), for all integersk,l.

Ifd is a positive integer, then

D(ad,bd;cd) =dD(a,b;c),

D(ad,bd;c) =D(a,b;c), if (d,c) = 1,

D(ad,b;cd) =D(a,b;c), if (d,b) = 1.

There is a proof for the last equality making use of

${\displaystyle \sum _{n=1}^{c-1}\left(\left(\frac{n+x}{c}\right)\right)=((x)),\mathrm{\forall }x\in \mathbb{R}.}$

Furthermore,az = 1 (modc) impliesD(a,b;c) =D(1,bz;c).

Ifb andc arecoprime, we may writes(b,c) as

${\displaystyle s(b,c)=\frac{-1}{c}\sum _{\omega }\frac{1}{(1-{\omega }^b)(1-\omega )}+\frac{1}{4}-\frac{1}{4c},}$

where the sum extends over thec-throots of unity other than 1, i.e. over all${\displaystyle \omega }$ such that${\displaystyle {\omega }^c=1}$ and${\displaystyle \omega \ne 1}$.

Ifb, c > 0 are coprime, then

${\displaystyle s(b,c)=\frac{1}{4c}\sum _{n=1}^{c-1}\cot \left(\frac{\pi n}{c}\right)\cot \left(\frac{\pi nb}{c}\right).}$

Ifb andc are coprime positive integers then

${\displaystyle s(b,c)+s(c,b)=\frac{1}{12}\left(\frac{b}{c}+\frac{1}{bc}+\frac{c}{b}\right)-\frac{1}{4}.}$

Rewriting this as

${\displaystyle 12bc\left(s(b,c)+s(c,b)\right)=b^2+c^2-3bc+1,}$

it follows that the number 6c s(b,c) is an integer.

Ifk = (3,c) then

${\displaystyle 12bcs(c,b)=0\mathrm{mod}kc}$

and

${\displaystyle 12bcs(b,c)=b^2+1\mathrm{mod}kc.}$

A relation that is prominent in the theory of theDedekind eta function is the following. Letq = 3, 5, 7 or 13 and letn = 24/(q − 1). Then given integersa,b,c,d withad − bc = 1 (thus belonging to themodular group), withc chosen so thatc = kq for some integerk > 0, define

${\displaystyle \delta =s(a,c)-\frac{a+d}{12c}-s(a,k)+\frac{a+d}{12k}}$

Thennδ is aneven integer.

Hans Rademacher found the following generalization of the reciprocity law for Dedekind sums:^[1] Ifa,b, andc are pairwise coprime positive integers, then

${\displaystyle D(a,b;c)+D(b,c;a)+D(c,a;b)=\frac{1}{12}\frac{a^2+b^2+c^2}{abc}-\frac{1}{4}.}$

Hence, the above triple sum vanishesif and only if (a,b,c) is a Markov triple, i.e. a solution of theMarkov equation

${\displaystyle a^2+b^2+c^2=3abc.}$

• Tom M. Apostol,Modular functions and Dirichlet Series in Number Theory (1990), Springer-Verlag, New York.ISBN 0-387-97127-0(See chapter 3.)
• Matthias Beck and Sinai Robins,Dedekind sums: a discrete geometric viewpointArchived 2011-05-18 at theWayback Machine, (2005 or earlier)
• Matthias Beck and Sinai Robins:Computing the Continuous Discretely: Integer-Point Enumeration in Polyhedra, 2nd Ed., Springer, ISBN 978-1-4939-2969-6 (2015) # Chap.8 Dedekind Sums.
• Hans Rademacher andEmil Grosswald,Dedekind Sums, Carus Math. Monographs, No.16, Math. Assoc. Amer., 1972.ISBN 0-88385-016-8.

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