Fungrim entry: 375afe

\left|\pi\!\left(x\right) - \operatorname{li}\!\left(x\right)\right| \lt \frac{\sqrt{x} \log\!\left(x\right)}{8 \pi}

Assumptions:

x \in \mathbb{R} \,\mathbin{\operatorname{and}}\, x \ge 2657 \,\mathbin{\operatorname{and}}\, \operatorname{RiemannHypothesis}

References:

L. Schoenfeld (1976). Sharper bounds for the Chebyshev functions θ(x) and ψ(x). II. Mathematics of Computation. 30 (134): 337-360. DOI: 10.2307/2005976

TeX:

\left|\pi\!\left(x\right) - \operatorname{li}\!\left(x\right)\right| \lt \frac{\sqrt{x} \log\!\left(x\right)}{8 \pi}

x \in \mathbb{R} \,\mathbin{\operatorname{and}}\, x \ge 2657 \,\mathbin{\operatorname{and}}\, \operatorname{RiemannHypothesis}

Definitions:

Fungrim symbol	Notation	Short description
`Abs`	$\left\|z\right\|$	Absolute value
`PrimePi`	$\pi\!\left(x\right)$	Prime counting function
`LogIntegral`	$\operatorname{li}\!\left(z\right)$	Logarithmic integral
`Sqrt`	$\sqrt{z}$	Principal square root
`Log`	$\log\!\left(z\right)$	Natural logarithm
`ConstPi`	$\pi$	The constant pi (3.14...)
`RR`	$\mathbb{R}$	Real numbers
`RiemannHypothesis`	$\operatorname{RiemannHypothesis}$	Riemann hypothesis

Source code for this entry:

Entry(ID("375afe"),
    Formula(Less(Abs(Sub(PrimePi(x), LogIntegral(x))), Div(Mul(Sqrt(x), Log(x)), Mul(8, ConstPi)))),
    Variables(x),
    Assumptions(And(Element(x, RR), GreaterEqual(x, 2657), RiemannHypothesis)),
    References("L. Schoenfeld (1976). Sharper bounds for the Chebyshev functions θ(x) and ψ(x). II. Mathematics of Computation. 30 (134): 337-360. DOI: 10.2307/2005976"))

Topics using this entry

Prime numbers