Given relatively prime integers  and  (i.e., ), the Dedekind sum is defined by

(1)

where

{x-|_x_|-1/2 x not in Z; 0 x in Z, " src="https://mathworld.wolfram.com/images/equations/DedekindSum/NumberedEquation2.gif" style="height:48px; width:166px" />

(2)

with  the floor function.  is an odd function since  and is periodic with period 1. The Dedekind sum is meaningful even if , so the relatively prime restriction is sometimes dropped (Apostol 1997, p. 72). The symbol  is sometimes used instead of  (Beck 2000).

The Dedekind sum can also be expressed in the form

(3)

If , let , , ...,  denote the remainders in the Euclidean algorithm given by

			(4)
			(5)
			(6)

for  and . Then

{(-1)^(j+1)(r_j^2+r_(j-1)^2+1)/(r_jr_(j-1))}-((-1)^n+1)/8 " src="https://mathworld.wolfram.com/images/equations/DedekindSum/NumberedEquation4.gif" style="height:50px; width:307px" />

(7)

(Apostol 1997, pp. 72-73).

In general, there is no simple formula for closed-form evaluation of , but some special cases are

			(8)
			(9)

(Apostol 1997, p. 62). Apostol (1997, p. 73) gives the additional special cases

(10)

(11)

(12)

(13)

for  and , where  and . Finally,

(14)

for  and , where  or .

Dedekind sums obey 2-term

(15)

(Dedekind 1953; Rademacher and Grosswald 1972; Pommersheim 1993; Apostol 1997, pp. 62-64) and 3-term

(16)

(Rademacher 1954), reciprocity laws, where , ; , ; and ,  are pairwise relatively prime, and

	(17)
	(18)
	(19)

(Pommersheim 1993).

 is an integer (Rademacher and Grosswald 1972, p. 28), and if , then

(20)

and

(21)

In addition,  satisfies the congruence

(22)

which, if  is odd, becomes

(23)

(Apostol 1997, pp. 65-66). If , 5, 7, or 13, let , let integers , , ,  be given with  such that  and , and let

{s(a,c)-(a+d)/(12c)}-{s(a,c_1)-(a+d)/(12c_1)}. " src="https://mathworld.wolfram.com/images/equations/DedekindSum/NumberedEquation16.gif" style="height:38px; width:256px" />

(24)

Then  is an even integer (Apostol 1997, pp. 66-69).

Let , , ,  with  (i.e., are pairwise relatively prime), then the Dedekind sums also satisfy

(25)

where , and ,  are any integers such that  (Pommersheim 1993).

If  is prime, then

(26)

(Dedekind 1953; Apostol 1997, p. 73). Moreover, it has been beautifully generalized by Knopp (1980).

REFERENCES:

Apostol, T. M. "Properties of Dedekind Sums," "The Reciprocity Law for Dedekind Sums," and "Congruence Properties of Dedekind Sums." §3.7-3.9 in Modular Functions and Dirichlet Series in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 52 and 61-69, 1997.

Apostol, T. M. Ch. 12 in Introduction to Analytic Number Theory. New York: Springer-Verlag, 1976.

Beck, M. "Dedekind Cotangent Sums" 7 Dec 2001. https://arxiv.org/abs/math.NT/0112077.

Dedekind, R. "Erlauterungen zu den Fragmenten, XXVIII." In The Collected Works of Bernhard Riemann. New York: Dover, pp. 466-478, 1953.

Iseki, S. "The Transformation Formula for the Dedekind Modular Function and Related Functional Equations." Duke Math. J. 24, 653-662, 1957.

Knopp, M. I. "Hecke Operators and an Identity for Dedekind Sums." J. Number Th. 12, 2-9, 1980.

Pommersheim, J. "Toric Varieties, Lattice Points, and Dedekind Sums." Math. Ann. 295, 1-24, 1993.

Rademacher, H. "Generalization of the Reciprocity Formula for Dedekind Sums." Duke Math. J. 21, 391-398, 1954.

Rademacher, H. and Grosswald, E. Dedekind Sums. Washington, DC: Math. Assoc. Amer., 1972.

Rademacher, H. and Whiteman, A. L. "Theorems on Dedekind Sums." Amer. J. Math. 63, 377-407, 1941.