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Fungrim entry: 8a9884

π4q=1φ(q)q4=π4ζ ⁣(3)ζ ⁣(4)=45ζ ⁣(3)2π3\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}}
\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}}
Fungrim symbol Notation Short description
Piπ\pi The constant pi (3.14...)
Sumnf(n)\sum_{n} f(n) Sum
Totientφ(n)\varphi(n) Euler totient function
Powab{a}^{b} Power
Infinity\infty Positive infinity
RiemannZetaζ ⁣(s)\zeta\!\left(s\right) Riemann zeta function
Source code for this entry:
    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))))))

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2021-03-15 19:12:00.328586 UTC