Fungrim entry: 23961e

s\!\left(n, k\right) = \sum_{r=1}^{k - 1} \frac{r}{k} \left(\frac{n r}{k} - \left\lfloor \frac{n r}{k} \right\rfloor - \frac{1}{2}\right)

Assumptions:

n \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; k \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; k > 0 \;\mathbin{\operatorname{and}}\; \gcd\!\left(n, k\right) = 1

TeX:

s\!\left(n, k\right) = \sum_{r=1}^{k - 1} \frac{r}{k} \left(\frac{n r}{k} - \left\lfloor \frac{n r}{k} \right\rfloor - \frac{1}{2}\right)

n \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; k \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; k > 0 \;\mathbin{\operatorname{and}}\; \gcd\!\left(n, k\right) = 1

Definitions:

Fungrim symbol	Notation	Short description
`DedekindSum`	$s\!\left(n, k\right)$	Dedekind sum
`Sum`	$\sum_{n} f(n)$	Sum
`ZZ`	$\mathbb{Z}$	Integers
`GCD`	$\gcd\!\left(a, b\right)$	Greatest common divisor

Source code for this entry:

Entry(ID("23961e"),
    Formula(Equal(DedekindSum(n, k), Sum(Mul(Div(r, k), Sub(Sub(Div(Mul(n, r), k), Floor(Div(Mul(n, r), k))), Div(1, 2))), For(r, 1, Sub(k, 1))))),
    Variables(n, k),
    Assumptions(And(Element(n, ZZ), Element(k, ZZ), Greater(k, 0), Equal(GCD(n, k), 1))))

Topics using this entry

Dedekind eta function