Dedekind sum

﻿
Dedekind sum

In mathematics, Dedekind sums, named after Richard Dedekind, 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 obey a large number of relationships on themselves; this article lists only a tiny fraction of these.

Definition

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

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

We then let

D :Z3R

be defined by

$D(a,b;c)=\sum_{n \bmod c} \left( \Bigg( \frac{an}{c} \Bigg) \right) \left( \left( \frac{bn}{c} \right) \right),$

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

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;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

$\sum_{n \bmod c} \left( \left( \frac{n+x}{c} \right) \right)=\left(\left( x\right)\right),\qquad\forall x\in\mathbb{R}.$

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

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

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

If b, c > 0 are coprime, then

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

Reciprocity law

If b and c are coprime positive integers then

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

Rewriting this as

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

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

If k = (3, c) then

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

and

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

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

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

Then one has nδ is an even integer.

Rademacher's generalization of the reciprocity law

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

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

References

1. ^ H. Rademacher, Generalization of the Reciprocity Formula for Dedekind Sums, Duke Mathematical Journal 21 (1954), pp. 391-397

Wikimedia Foundation. 2010.

Look at other dictionaries:

• Dedekind eta function — For the Dirichlet series see Dirichlet eta function. Dedekind η function in the complex plane The Dedekind eta function, named after Richard Dedekind, is a function defined on the upper half plane of complex numbers, where the imaginary part is… …   Wikipedia

• Dedekind domain — In abstract algebra, a Dedekind domain or Dedekind ring, named after Richard Dedekind, is an integral domain in which every nonzero proper ideal factors into a product of prime ideals. It can be shown that such a factorization is then necessarily …   Wikipedia

• Dedekind, Richard — ▪ German mathematician born Oct. 6, 1831, Braunschweig, duchy of Braunschweig [Germany] died Feb. 12, 1916, Braunschweig  German mathematician who developed a major redefinition of irrational numbers (irrational number) in terms of arithmetic… …   Universalium

• Dedekind zeta function — In mathematics, the Dedekind zeta function of an algebraic number field K, generally denoted ζK(s), is a generalization of the Riemann zeta function which is obtained by specializing to the case where K is the rational numbers Q. In particular,… …   Wikipedia

• Dedekind psi function — In number theory, the Dedekind psi function is the multiplicative function on the positive integers defined by where the product is taken over all primes p dividing n (by convention, ψ(1) is the empty product and so has value 1). The function was …   Wikipedia

• Richard Dedekind — Infobox Scientist name = PAGENAME box width = image size =180px caption =Richard Dedekind, c. 1850 birth date = October 6, 1831 birth place = Braunschweig death date = February 12, 1916 death place = Braunschweig residence = citizenship =… …   Wikipedia

• Fonction Zeta De Dedekind — Fonction zêta de Dedekind En mathématiques, la fonction zêta de Dedekind est une série de Dirichlet définie pour tout corps de nombres , et notée où s est une variable complexe. C est la somme infinie prise sur tous les idéaux I de l anneau des… …   Wikipédia en Français

• Función zeta de Dedekind — En matemática, la función zeta de Dedekind es una serie de Dirichlet definida para todo cuerpo K de números algebraicos, expresada como ζK(s) donde s es una variable compleja. Es la suma infinita: realizada sobre todos los I ideales del anillo de …   Wikipedia Español

• Fonction zêta de Dedekind — En mathématiques, la fonction zêta de Dedekind est une série de Dirichlet définie pour tout corps de nombres , et notée où s est une variable complexe. C est la somme infinie prise sur tous les idéaux I de l anneau des entiers de K, avec …   Wikipédia en Français

• Ramanujan's sum — This article is not about Ramanujan summation. In number theory, a branch of mathematics, Ramanujan s sum, usually denoted c q ( n ), is a function of two positive integer variables q and n defined by the formula:c q(n)=sum {a=1atop… …   Wikipedia