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Fungrim entry: 8d7b3d

lim⁡N→∞1N2∑n=1Nφ(n)=3π2\lim_{N \to \infty} \frac{1}{{N}^{2}} \sum_{n=1}^{N} \varphi(n) = \frac{3}{{\pi}^{2}}
TeX:
\lim_{N \to \infty} \frac{1}{{N}^{2}} \sum_{n=1}^{N} \varphi(n) = \frac{3}{{\pi}^{2}}
Definitions:
Fungrim symbol Notation Short description
SequenceLimitlim⁡n→af(n)\lim_{n \to a} f(n) Limiting value of sequence
Powab{a}^{b} Power
Sum∑nf(n)\sum_{n} f(n) Sum
Totientφ(n)\varphi(n) Euler totient function
Infinity∞\infty Positive infinity
Piπ\pi The constant pi (3.14...)
Source code for this entry:
Entry(ID("8d7b3d"),
    Formula(Equal(SequenceLimit(Mul(Div(1, Pow(N, 2)), Sum(Totient(n), For(n, 1, N))), For(N, Infinity)), Div(3, Pow(Pi, 2)))))

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

2021-03-15 19:12:00.328586 UTC