Fungrim entry: 4099d2

\lim_{N \to \infty} \frac{1}{{N}^{n}} \left|\left\{ T : T \in {\left(\{1, 2, \ldots N\}\right)}^{n} \,\mathbin{\operatorname{and}}\, \gcd\!\left(T\right) = 1 \right\}\right| = \frac{1}{\zeta\!\left(n\right)}

Assumptions:

n \in \mathbb{Z}_{\ge 2}

TeX:

\lim_{N \to \infty} \frac{1}{{N}^{n}} \left|\left\{ T : T \in {\left(\{1, 2, \ldots N\}\right)}^{n} \,\mathbin{\operatorname{and}}\, \gcd\!\left(T\right) = 1 \right\}\right| = \frac{1}{\zeta\!\left(n\right)}

n \in \mathbb{Z}_{\ge 2}

Definitions:

Fungrim symbol	Notation	Short description
`SequenceLimit`	$\lim_{n \to a} f\!\left(n\right)$	Limiting value of sequence
`Pow`	${a}^{b}$	Power
`Cardinality`	$\left\|S\right\|$	Set cardinality
`SetBuilder`	$\left\{ f\!\left(x\right) : P\!\left(x\right) \right\}$	Set comprehension
`ZZBetween`	$\{a, a + 1, \ldots b\}$	Integers between a and b inclusive
`GCD`	$\gcd\!\left(n, k\right)$	Greatest common divisor
`Infinity`	$\infty$	Positive infinity
`RiemannZeta`	$\zeta\!\left(s\right)$	Riemann zeta function
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("4099d2"),
    Formula(Equal(SequenceLimit(Mul(Div(1, Pow(N, n)), Cardinality(SetBuilder(T, T, And(Element(T, Pow(ZZBetween(1, N), n)), Equal(GCD(T), 1))))), N, Infinity), Div(1, RiemannZeta(n)))),
    Variables(n),
    Assumptions(Element(n, ZZGreaterEqual(2))))

Topics using this entry

Greatest common divisor