Given relatively prime integers and (i.e., ), the Dedekind sum is defined by
(1)
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where
(2)
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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)
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If , let , , ..., denote the remainders in the Euclidean algorithm given by
(4)
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(5)
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(6)
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for and . Then
(7)
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(Apostol 1997, pp. 72-73).
In general, there is no simple formula for closed-form evaluation of , but some special cases are
(8)
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(9)
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(Apostol 1997, p. 62). Apostol (1997, p. 73) gives the additional special cases
(10)
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(11)
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(12)
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(13)
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for and , where and . Finally,
(14)
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for and , where or .
Dedekind sums obey 2-term
(15)
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(Dedekind 1953; Rademacher and Grosswald 1972; Pommersheim 1993; Apostol 1997, pp. 62-64) and 3-term
(16)
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(Rademacher 1954), reciprocity laws, where , ; , ; and , are pairwise relatively prime, and
(17)
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(18)
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(19)
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(Pommersheim 1993).
is an integer (Rademacher and Grosswald 1972, p. 28), and if , then
(20)
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and
(21)
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In addition, satisfies the congruence
(22)
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which, if is odd, becomes
(23)
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(Apostol 1997, pp. 65-66). If , 5, 7, or 13, let , let integers , , , be given with such that and , and let
(24)
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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)
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where , and , are any integers such that (Pommersheim 1993).
If is prime, then
(26)
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(Dedekind 1953; Apostol 1997, p. 73). Moreover, it has been beautifully generalized by Knopp (1980).