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Fungrim entry: 0477b3

φ(n)σ1 ⁣(n)>6π2n2\varphi(n) \sigma_{1}\!\left(n\right) > \frac{6}{{\pi}^{2}} {n}^{2}
Assumptions:nZ1n \in \mathbb{Z}_{\ge 1}
References:
  • G. H. Hardy and E. M. Wright (1979), An Introduction to the Theory of Numbers (Fifth ed.), Oxford University Press. Theorem 327.
TeX:
\varphi(n) \sigma_{1}\!\left(n\right) > \frac{6}{{\pi}^{2}} {n}^{2}

n \in \mathbb{Z}_{\ge 1}
Definitions:
Fungrim symbol Notation Short description
Totientφ(n)\varphi(n) Euler totient function
DivisorSigmaσk ⁣(n)\sigma_{k}\!\left(n\right) Sum of divisors function
Powab{a}^{b} Power
ConstPiπ\pi The constant pi (3.14...)
ZZGreaterEqualZn\mathbb{Z}_{\ge n} Integers greater than or equal to n
Source code for this entry:
Entry(ID("0477b3"),
    Formula(Greater(Mul(Totient(n), DivisorSigma(1, n)), Mul(Div(6, Pow(ConstPi, 2)), Pow(n, 2)))),
    Variables(n),
    Assumptions(Element(n, ZZGreaterEqual(1))),
    References("G. H. Hardy and E. M. Wright (1979), An Introduction to the Theory of Numbers (Fifth ed.), Oxford University Press. Theorem 327."))

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2019-10-05 13:11:19.856591 UTC