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Fungrim entry: 23961e

s ⁣(n,k)=r=1k1rk(nrknrk12)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:nZandkZandk>0andgcd ⁣(n,k)=1n \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
DedekindSums ⁣(n,k)s\!\left(n, k\right) Dedekind sum
Sumnf(n)\sum_{n} f(n) Sum
ZZZ\mathbb{Z} Integers
GCDgcd ⁣(a,b)\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))))

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

2019-11-19 15:10:20.037976 UTC