Fungrim entry: 8a9884

\frac{\pi}{4} \sum_{q=1}^{\infty} \frac{\varphi(q)}{{q}^{4}} = \frac{\pi}{4} \frac{\zeta\!\left(3\right)}{\zeta\!\left(4\right)} = \frac{45 \zeta\!\left(3\right)}{2 {\pi}^{3}}

TeX:

\frac{\pi}{4} \sum_{q=1}^{\infty} \frac{\varphi(q)}{{q}^{4}} = \frac{\pi}{4} \frac{\zeta\!\left(3\right)}{\zeta\!\left(4\right)} = \frac{45 \zeta\!\left(3\right)}{2 {\pi}^{3}}

Definitions:

Fungrim symbol	Notation	Short description
`Pi`	$\pi$	The constant pi (3.14...)
`Sum`	$\sum_{n} f(n)$	Sum
`Totient`	$\varphi(n)$	Euler totient function
`Pow`	${a}^{b}$	Power
`Infinity`	$\infty$	Positive infinity
`RiemannZeta`	$\zeta\!\left(s\right)$	Riemann zeta function

Source code for this entry:

Entry(ID("8a9884"),
    Formula(Equal(Mul(Div(Pi, 4), Sum(Div(Totient(q), Pow(q, 4)), For(q, 1, Infinity))), Mul(Div(Pi, 4), Div(RiemannZeta(3), RiemannZeta(4))), Div(Mul(45, RiemannZeta(3)), Mul(2, Pow(Pi, 3))))))

Topics using this entry

Modular transformations

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