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In mathematics, Dedekind sums, named after Richard Dedekind, are certain sums of products of a sawtooth function s, 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 as x-lfloor x floor - 1/2, &mboxxinmathbbsetminusmathbb;\ 0,&mboxxinmathbb. end We then let DZ3 → R be defined by ight) ight) left( left( rac ight) ight), 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 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 ight) ight)=left(left( x ight) ight),qquad orall xinmathbb. 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 rac + rac - rac, 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 cot left( rac ight) cot left( rac ight). Reciprocity law If b and c are coprime positive integers then ight)- rac. Rewriting this as ight) = b^2 + c^2 -3bc + 1, 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 one has nδ is an even integer. Rademachers generalization of the reciprocity law Hans Rademacher found the following generalization of the reciprocity law for Dedekind sums: If a,b, and c are pairwise coprime positive integers, then | ||||||||
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