In mathematics, Dedekind sums, named after Richard Dedekind, are certain sums of products of a sawtooth function s, and are given by a function D of three integer variables. Dedekind introduced them to express the functional equation of the Dedekind eta function. They have subsequently been much studied in number theory, and have occurred in some problems of topology. Dedekind sums obey a large number of relationships on themselves; this article lists only a tiny fraction of these.

Definition

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Define the sawtooth function $\backslash left(\; \backslash left(\; \backslash right)\; \backslash right):\backslash mathbb\{R\}\; \backslash rightarrow\; \backslash mathbb\{R\}$ as

$((x))=\backslash begin\{cases\}$

x-\lfloor x\rfloor - 1/2, &\mbox{if }x\in\mathbb{R}\setminus\mathbb{Z};\\ 0,&\mbox{if }x\in\mathbb{Z}. \end{cases}

We then let

D :Z³ → R

be defined by

$D(a,b;c)=\backslash sum\_\{n\backslash mod\; c\}\; \backslash left(\; \backslash left(\; \backslash frac\{an\}\{c\}\; \backslash right)\; \backslash right)\; \backslash left(\; \backslash left(\; \backslash frac\{bn\}\{c\}\; \backslash right)\; \backslash right),$

the terms on the right being the Dedekind sums. For the case a=1, one often writes

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

Simple formulae

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Note that D is symmetric in a and b, and hence

$D(a,b;c)=D(b,a;c),$

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 of D 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 integers k,l.

If d 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

$\backslash sum\_\{n\backslash mod\; c\}\; \backslash left(\; \backslash left(\; \backslash frac\{n+x\}\{c\}\; \backslash right)\; \backslash right)=\backslash left(\backslash left(\; x\backslash right)\backslash right),\backslash qquad\backslash forall\; x\backslash in\backslash mathbb\{R\}.$

Furthermore, az = 1 (mod c) implies D(a,b;c) = D(1,bz;c).

Alternative forms

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If b and c are coprime, we may write s(b,c) as

$s(b,c)=\backslash frac\{-1\}\{c\}\; \backslash sum\_\backslash omega$

\frac{1} { (1-\omega^b) (1-\omega ) } +\frac{1}{4} - \frac{1}{4c},

where the sum extends over the c-th roots of unity other than 1, i.e. over all $\backslash omega$ such that $\backslash omega^c=1$ and $\backslash omega\backslash not=1$.

If b, c > 0 are coprime, then

$s(b,c)=\backslash frac\{1\}\{4c\}\backslash sum\_\{n=1\}^\{c-1\}$

\cot \left( \frac{\pi n}{c} \right) \cot \left( \frac{\pi nb}{c} \right).

Reciprocity law

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If b and c are coprime positive integers then

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

Rewriting this as

$12bc\; \backslash left(\; s(b,c)\; +\; s(c,b)\; \backslash right)\; =\; b^2\; +\; c^2\; -3bc\; +\; 1,$

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

If k = (3, c) then

$12bc\backslash ,\; s(c,b)=0\; \backslash mod\; kc$

and

$12bc\backslash ,\; s(b,c)=b^2+1\; \backslash mod\; kc.$

A relation that is prominent in the theory of the Dedekind eta function is the following. Let q = 3, 5, 7 or 13 and let n = 24/(q − 1). Then given integers a, b, c, d with ad − bc = 1 (thus belonging to the modular group), with c chosen so that c = kq for some integer k > 0, define

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

Then one has nδ is an even integer.

Rademacher's generalization of the reciprocity law

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Hans Rademacher found the following generalization of the reciprocity law for Dedekind sums:H. Rademacher, Generalization of the Reciprocity Formula for Dedekind Sums, Duke Mathematical Journal 21 (1954), pp. 391-397 If a,b, and c'' are pairwise coprime positive integers, then

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

References

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• 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 viewpoint, (2005 or earlier)
• H. Rademacher and Emil Grosswald, Dedekind Sums, Carus Math. Monographs, 1972

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