Dedekind sum Last updated July 08, 2025 Definition Define the sawtooth function ( ( ) ) : R → R {\displaystyle (\!(\,)\!):\mathbb {R} \rightarrow \mathbb {R} } as
( ( x ) ) = { x − ⌊ x ⌋ − 1 / 2 , if x ∈ R ∖ Z ; 0 , if x ∈ Z . {\displaystyle (\!(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 : Z 2 × ( Z − { 0 } ) → R {\displaystyle D:\mathbb {Z} ^{2}\times (\mathbb {Z} -\{0\})\to \mathbb {R} } be defined by
D ( a , b ; c ) = ∑ n = 1 c − 1 ( ( a n c ) ) ( ( b n c ) ) , {\displaystyle D(a,b;c)=\sum _{n=1}^{c-1}\left(\!\!\left({\frac {an}{c}}\right)\!\!\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 ).Note that D is symmetric in a and b , and hence
D ( a , b ; c ) = D ( b , a ; c ) , {\displaystyle 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
∑ n = 1 c − 1 ( ( n + x c ) ) = ( ( x ) ) , ∀ x ∈ R . {\displaystyle \sum _{n=1}^{c-1}\left(\!\!\left({\frac {n+x}{c}}\right)\!\!\right)=(\!(x)\!),\qquad \forall x\in \mathbb {R} .} Furthermore, az = 1 (mod c ) implies D (a , b ; c ) = D (1, bz ; c ).
If b and c are coprime , we may write s (b , c ) as
s ( b , c ) = − 1 c ∑ ω 1 ( 1 − ω b ) ( 1 − ω ) + 1 4 − 1 4 c , {\displaystyle 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 {\displaystyle \omega } such that ω c = 1 {\displaystyle \omega ^{c}=1} and ω ≠ 1 {\displaystyle \omega \not =1} .
If b , c > 0 are coprime, then
s ( b , c ) = 1 4 c ∑ n = 1 c − 1 cot ( π n c ) cot ( π n b c ) . {\displaystyle 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 ) = 1 12 ( b c + 1 b c + c b ) − 1 4 . {\displaystyle 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
12 b c ( s ( b , c ) + s ( c , b ) ) = b 2 + c 2 − 3 b c + 1 , {\displaystyle 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
12 b c s ( c , b ) = 0 mod k c {\displaystyle 12bc\,s(c,b)=0\mod kc} and
12 b c s ( b , c ) = b 2 + 1 mod k c . {\displaystyle 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
δ = s ( a , c ) − a + d 12 c − s ( a , k ) + a + d 12 k {\displaystyle \delta =s(a,c)-{\frac {a+d}{12c}}-s(a,k)+{\frac {a+d}{12k}}} Then n δ 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 ) = 1 12 a 2 + b 2 + c 2 a b c − 1 4 . {\displaystyle 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}}.} 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
a 2 + b 2 + c 2 = 3 a b c . {\displaystyle a^{2}+b^{2}+c^{2}=3abc.} 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.) Matthias Beck and Sinai Robins, Dedekind sums: a discrete geometric viewpoint Archived 2011-05-18 at the Wayback Machine , (2005 or earlier) Matthias Beck and Sinai Robins: Computing the Continuous Discretely: Integer-Point Enumeration in Polyhedra , 2nd Ed., Springer, ISBN 978-1-4939-2969-6 (2015) # Chap.8 Dedekind Sums. Hans Rademacher and Emil Grosswald , Dedekind Sums , Carus Math. Monographs, No.16, Math. Assoc. Amer., 1972. ISBN 0-88385-016-8 . This page is based on this
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