Dedekind sum

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In mathematics, Dedekind sums are certain sums of products of a sawtooth function, 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 have a large number of functional equations; this article lists only a small fraction of these.

Dedekind sums were introduced by Richard Dedekind in a commentary on fragment XXVIII of Bernhard Riemann's collected papers.

Definition

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Define the sawtooth function   as

 

We then let

 

be defined by

 

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

 

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

 

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

 

where the sum extends over the c-th roots of unity other than 1, i.e. over all   such that   and  .

If b, c > 0 are coprime, then

 

Reciprocity law

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If b and c are coprime positive integers then

 

Rewriting this as

 

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

If k = (3, c) then

 

and

 

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

 

Then 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:[1] If a, b, and c are pairwise coprime positive integers, then

 

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

 

References

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  1. ^ Rademacher, Hans (1954). "Generalization of the reciprocity formula for Dedekind sums". Duke Mathematical Journal. 21: 391–397. doi:10.1215/s0012-7094-54-02140-7. Zbl 0057.03801.

Further reading

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