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

Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2020-08-27 09:56:25.682319 UTC