Dedekind sum

In mathematics, Dedekind sums are certain sums of products of a sawtooth function s, and are given by a function D of three integer variables. They are named after the mathematician Richard Dedekind, who 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

Define the sawtooth function <math>\left( \left( \right) \right):\mathbb{R} \rightarrow \mathbb{R}<math> as

<math>((x))=\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}<math>.

Then the function

D :Z³ → R

defined by

<math>D(a,b;c)=\sum_{n\mod c} \left( \left( \frac{an}{c} \right) \right) \left( \left( \frac{bn}{c} \right) \right)<math>

is called a Dedekind sum. For the case a=1, one often writes

<math>s(b,c)=D(1,b;c)<math>

Alternate forms

For integers b > 0 and c > 0, one can also write

<math>s(b,c)=\sum_{n=1}^{c-1} \frac{n}{c}

\left( \left( \frac {nb}{c} \right) \right) <math> and

<math>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) <math> and

<math>s(b,c)=\frac{-1}{c} \sum_{\omega^c=1}

\frac{1} { (1-\omega^b) (1-\omega ) } +\frac{1}{4} – \frac{1}{4c} <math>

where the sum extends over <math>\omega<math> the c 'th root of unity.

Properties

Note that for c > 0,

<math>\sum_{n \mod c} \left( \left( \frac {n}{c} \right) \right) =0<math>

and more generally,

<math>\sum_{n \mod c} \left( \left( \frac {nb}{c} \right) \right) =0<math>

for b coprime to c, that is, (b,c)=1.

If <math>d=\pm b \mod c<math> then <math>s(d,c) = \pm s(b,c)<math> with the same sign being taken as in the congruence. Does this equation hold for general D(a,b;c) ??

If <math>bd=\pm 1 \mod c<math> then <math>s(d,c) = \pm s(b,c)<math>.

If <math>b^2+1=0 \mod c<math> then <math>s(b,c) = 0<math>.

Reciprocity law

If b > 0 and c > 0 and (b,c) = 1 then

<math>12bc \left( s(b,c) + s(c,b) \right) = b^2 + c^2 -3bc + 1<math>

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

If k = (3, c) then

<math> 12bc\, s(c,b)=0 \mod kc<math>

and

<math> 12bc\, s(b,c)=b^2+1 \mod kc<math>

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

<math>\delta = s(a,c) – \frac{a+d}{12c} – s(a,k) + \frac{a+d}{12k}<math>

Then one has nδ is an even integer.

References

• 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) [1]