DLMF:27.12.E5 (Q8089)

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DLMF:27.12.E5
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    Statements

    Defining formula
    | π ⁡ ( x ) - li ⁡ ( x ) | = O ⁡ ( x ⁢ exp ⁡ ( - c ⁢ ( ln ⁡ x ) 1 / 2 ) ) , number-of-primes-less-pi 𝑥 logarithmic-integral 𝑥 Big-O 𝑥 𝑐 superscript 𝑥 1 2 {\displaystyle{\displaystyle\left|\pi\left(x\right)-\mathrm{li}\left(x\right)% \right|=O\left(x\exp\left(-c(\ln x)^{1/2}\right)\right),}}
    0 references
    DLMF id
    DLMF:27.12.E5
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    constraint
    x → ∞ → 𝑥 {\displaystyle{\displaystyle x\to\infty}}
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    Symbols used
    order not exceeding
    Defining formula
    O ⁡ ( x ) Big-O 𝑥 {\displaystyle{\displaystyle O\left(\NVar{x}\right)}}
    xml-id
    C2.S1.E3.m2adec
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    exponential function
    Defining formula
    exp ⁡ z 𝑧 {\displaystyle{\displaystyle\exp\NVar{z}}}
    xml-id
    C4.S2.E19.m2adec
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    logarithmic integral
    Defining formula
    li ⁡ ( x ) logarithmic-integral 𝑥 {\displaystyle{\displaystyle\mathrm{li}\left(\NVar{x}\right)}}
    xml-id
    C6.S2.E8.m2adec
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    principal branch of logarithm function
    Defining formula
    ln ⁡ z 𝑧 {\displaystyle{\displaystyle\ln\NVar{z}}}
    xml-id
    C4.S2.E2.m2acdec
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    number of primes not exceeding $$x$$
    Defining formula
    π ⁡ ( x ) number-of-primes-less-pi 𝑥 {\displaystyle{\displaystyle\pi\left(\NVar{x}\right)}}
    xml-id
    C27.S2.E2.m2abdec
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    Q12237
    Defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C27.S1.XMD6.m1bdec
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