Fungrim entry: 375afe

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

Assumptions:

x \in \mathbb{R} \;\mathbin{\operatorname{and}}\; x \ge 2657 \;\mathbin{\operatorname{and}}\; \operatorname{RH}

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

x \in \mathbb{R} \;\mathbin{\operatorname{and}}\; x \ge 2657 \;\mathbin{\operatorname{and}}\; \operatorname{RH}

Definitions:

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

Fungrim entry: 375afe

Topics using this entry