Fungrim entry: c2e919

s\!\left(n, k\right) = \sum_{r=1}^{k - 1} Q\!\left(\frac{r}{k}\right) Q\!\left(\frac{n r}{k}\right)\; \text{ where } Q(x) = \begin{cases} x - \left\lfloor x \right\rfloor - \frac{1}{2}, & x \notin \mathbb{Z}\\0, & x \in \mathbb{Z}\\ \end{cases}

Assumptions:

n \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; k \in \mathbb{Z}

TeX:

s\!\left(n, k\right) = \sum_{r=1}^{k - 1} Q\!\left(\frac{r}{k}\right) Q\!\left(\frac{n r}{k}\right)\; \text{ where } Q(x) = \begin{cases} x - \left\lfloor x \right\rfloor - \frac{1}{2}, & x \notin \mathbb{Z}\\0, & x \in \mathbb{Z}\\ \end{cases}

n \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; k \in \mathbb{Z}

Definitions:

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

Source code for this entry:

Entry(ID("c2e919"),
    Formula(Equal(DedekindSum(n, k), Where(Sum(Mul(Q(Div(r, k)), Q(Div(Mul(n, r), k))), For(r, 1, Sub(k, 1))), Def(Q(x), Cases(Tuple(Sub(Sub(x, Floor(x)), Div(1, 2)), NotElement(x, ZZ)), Tuple(0, Element(x, ZZ))))))),
    Variables(n, k),
    Assumptions(And(Element(n, ZZ), Element(k, ZZ))))

Topics using this entry

Dedekind eta function