# Fungrim entry: 4099d2

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

n \in \mathbb{Z}_{\ge 2}
Definitions:
Fungrim symbol Notation Short description
SequenceLimit$\lim_{n \to a} f\!\left(n\right)$ Limiting value of sequence
Pow${a}^{b}$ Power
Cardinality$\# S$ 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$\gcd\!\left(a, b\right)$ 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(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

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

2019-08-21 11:44:15.926409 UTC