Dedekind sum
In mathematics, Dedekind sums are certain finite sums of products of a sawtooth function. Dedekind introduced them in the 1880's to express the functional equation of the Dedekind eta function, in a commentary to Bernhard Riemann's collected papers.[1]. They have subsequently been much studied in number theory but also occur in some results in topology,[2] geometric combinatorics,[3] algebraic geometry,[4] and computational complexity.[5] Dedekind sums have been generalized in various directions, satisfying a large number of functional equations; this article lists only a small fraction of these.
Definition
Define the sawtooth function as
We then define the Dedekind sum
by
For the case a = 1, one often writes
- s(b, c) = D(1, b; c).
Simple formulae
Note that D is symmetric in a and b, i.e.,
and that, by the oddness of (( )),
- 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
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 .[6]
Equivalently, if b and c are coprime, then
This reformulation mirrors the fact that the above cotangent function is the discrete Fourier transform of the sawtooth function.[6]
The reciprocity law
Dedekind[1] proved that, if b and c are coprime positive integers then
There exist several proofs from first principles, and Dedekind's reciprocity law is equivalent to quadratic reciprocity.[7]
Rewriting the reciprocity law 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 is an even integer.
Rademacher's generalization of the reciprocity law
Hans Rademacher found the following generalization of the reciprocity law for Dedekind sums:[8] 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
- ^ a b Dedekind, Richard (1953). "Erläuterungen zu den Fragmenten XXVIII". Collected Works of Bernhard Riemann. Dover Publ., New York. pp. 466–478.
- ^ Hirzebruch, Friedrich; Zagier, Don (1974). The Atiyah–Singer Theorem and Elementary Number Theory. Boston, Mass.: Publish or Perish.
- ^ Pommersheim, James E. (1993). "Toric varieties, lattice points and Dedekind sums". Math. Ann. 295 (1): 1–24.
- ^ Garoufalidis, Stavros; Pommersheim, James E. (2001). "Values of zeta functions at negative integers, Dedekind sums and toric geometry". J. Amer. Math. Soc. 14 (1): 1–23.
- ^ Knuth, Donald E. (1981). The Art of Computer Programming. Vol. 2. Reading, Mass.: Addison-Wesley Publishing Co.
- ^ a b Beck, Matthias; Robins, Sinai (2015). "Chapter 8. Dedekind Sums". Computing the Continuous Discretely: Integer-Point Enumeration in Polyhedra. New York: Springer. ISBN 978-1-4939-2969-6.
- ^ Rademacher, Hans; Grosswald, Emil (1972). Dedekind Sums. Math. Assoc. Amer. ISBN 0-88385-016-8.
- ^ 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
- Tom M. Apostol, Modular functions and Dirichlet Series in Number Theory (1990), Springer-Verlag, New York. ISBN 0-387-97127-0 (See chapter 3.)