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Fungrim entry: 375afe

π(x)li(x)<xlog(x)8π\left|\pi(x) - \operatorname{li}(x)\right| < \frac{\sqrt{x} \log(x)}{8 \pi}
Assumptions:xR  and  x2657  and  RHx \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
Absz\left|z\right| Absolute value
PrimePiπ(x)\pi(x) Prime counting function
LogIntegralli(z)\operatorname{li}(z) Logarithmic integral
Sqrtz\sqrt{z} Principal square root
Loglog(z)\log(z) Natural logarithm
Piπ\pi The constant pi (3.14...)
RRR\mathbb{R} Real numbers
RiemannHypothesisRH\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"))

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2020-04-08 16:14:44.404316 UTC