# 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

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