Fungrim entry: 4099d2

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

Assumptions:

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

TeX:

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

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

Definitions:

Fungrim symbol	Notation	Short description
`SequenceLimit`	$\lim_{n \to a} f(n)$	Limiting value of sequence
`Pow`	${a}^{b}$	Power
`Cardinality`	$\# S$	Set cardinality
`Range`	$\{a, a + 1, \ldots, b\}$	Integers between given endpoints
`GCD`	$\gcd\!\left(a, b\right)$	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(Set(T, For(T), And(Element(T, Pow(Range(1, N), n)), Equal(GCD(T), 1))))), For(N, Infinity)), Div(1, RiemannZeta(n)))),
    Variables(n),
    Assumptions(Element(n, ZZGreaterEqual(2))))

Topics using this entry

Greatest common divisor