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Fungrim entry: 375afe

π ⁣(x)li ⁣(x)<xlog ⁣(x)8π\left|\pi\!\left(x\right) - \operatorname{li}\!\left(x\right)\right| < \frac{\sqrt{x} \log\!\left(x\right)}{8 \pi}
Assumptions:xRandx2657andRHx \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\!\left(x\right) - \operatorname{li}\!\left(x\right)\right| < \frac{\sqrt{x} \log\!\left(x\right)}{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\!\left(x\right) Prime counting function
LogIntegralli ⁣(z)\operatorname{li}\!\left(z\right) Logarithmic integral
Sqrtz\sqrt{z} Principal square root
Loglog ⁣(z)\log\!\left(z\right) Natural logarithm
ConstPiπ\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, 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"))

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Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2019-08-17 11:32:46.829430 UTC