Given relatively prime integers 
and 
(i.e., 
), the Dedekind sum is defined by
 |
(1)
|
where
 |
(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
 |
(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)
|
![12hks(h,k)=(k-2)[k-1/2(h^2+1)] for k=2 (mod h)](/images/equations/DedekindSum/NumberedEquation6.svg) |
(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
 |
(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).
