Fungrim entry: 9923b7

\lim_{N \to \infty} \frac{1}{\log(N)} \sum_{n=1}^{N} \frac{1}{\varphi(n)} = \frac{\zeta\!\left(2\right) \zeta\!\left(3\right)}{\zeta\!\left(6\right)}

TeX:

\lim_{N \to \infty} \frac{1}{\log(N)} \sum_{n=1}^{N} \frac{1}{\varphi(n)} = \frac{\zeta\!\left(2\right) \zeta\!\left(3\right)}{\zeta\!\left(6\right)}

Definitions:

Fungrim symbol	Notation	Short description
`SequenceLimit`	$\lim_{n \to a} f(n)$	Limiting value of sequence
`Log`	$\log(z)$	Natural logarithm
`Sum`	$\sum_{n} f(n)$	Sum
`Totient`	$\varphi(n)$	Euler totient function
`Infinity`	$\infty$	Positive infinity
`RiemannZeta`	$\zeta\!\left(s\right)$	Riemann zeta function

Source code for this entry:

Entry(ID("9923b7"),
    Formula(Equal(SequenceLimit(Mul(Div(1, Log(N)), Sum(Div(1, Totient(n)), For(n, 1, N))), For(N, Infinity)), Div(Mul(RiemannZeta(2), RiemannZeta(3)), RiemannZeta(6)))))

Topics using this entry

Totient function

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