Fungrim entry: bfa464

p_{n} < n \left(\log(n) + \log\!\left(\log(n)\right) - 1 + \frac{\log\!\left(\log(n)\right) - 2}{\log(n)} - \frac{\log^{2}\!\left(\log(n)\right) - 6 \log\!\left(\log(n)\right) + 10.667}{2 \log^{2}\!\left(n\right)}\right)

Assumptions:

n \in \mathbb{Z}_{\ge 46254381}

References:

https://arxiv.org/abs/1706.03651

TeX:

p_{n} < n \left(\log(n) + \log\!\left(\log(n)\right) - 1 + \frac{\log\!\left(\log(n)\right) - 2}{\log(n)} - \frac{\log^{2}\!\left(\log(n)\right) - 6 \log\!\left(\log(n)\right) + 10.667}{2 \log^{2}\!\left(n\right)}\right)

n \in \mathbb{Z}_{\ge 46254381}

Definitions:

Fungrim symbol	Notation	Short description
`PrimeNumber`	$p_{n}$	nth prime number
`Log`	$\log(z)$	Natural logarithm
`Pow`	${a}^{b}$	Power
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("bfa464"),
    Formula(Less(PrimeNumber(n), Mul(n, Sub(Add(Sub(Add(Log(n), Log(Log(n))), 1), Div(Sub(Log(Log(n)), 2), Log(n))), Div(Add(Sub(Pow(Log(Log(n)), 2), Mul(6, Log(Log(n)))), Decimal("10.667")), Mul(2, Pow(Log(n), 2))))))),
    Variables(n),
    Assumptions(Element(n, ZZGreaterEqual(46254381))),
    References("https://arxiv.org/abs/1706.03651"))

Topics using this entry

Prime numbers