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Fungrim entry: 4099d2

limN1Nn#{T:T({1,2,N})nandgcd ⁣(T)=1}=1ζ ⁣(n)\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:nZ2n \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
SequenceLimitlimnaf ⁣(n)\lim_{n \to a} f\!\left(n\right) Limiting value of sequence
Powab{a}^{b} Power
Cardinality#S\# S Set cardinality
SetBuilder{f ⁣(x):P ⁣(x)}\left\{ f\!\left(x\right) : P\!\left(x\right) \right\} Set comprehension
ZZBetween{a,a+1,b}\{a, a + 1, \ldots b\} Integers between a and b inclusive
GCDgcd ⁣(a,b)\gcd\!\left(a, b\right) Greatest common divisor
Infinity\infty Positive infinity
RiemannZetaζ ⁣(s)\zeta\!\left(s\right) Riemann zeta function
ZZGreaterEqualZn\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))))

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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