DLMF:27.11.E12 (Q8081)

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DLMF:27.11.E12
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    Statements

    Defining formula
    ∑ n ≤ x μ ⁡ ( n ) = O ⁡ ( x ⁢ e - C ⁢ ln ⁡ x ) , subscript 𝑛 𝑥 Moebius-mu 𝑛 Big-O 𝑥 superscript 𝑒 𝐶 𝑥 {\displaystyle{\displaystyle\sum_{n\leq x}\mu\left(n\right)=O\left(xe^{-C\sqrt% {\ln x}}\right),}}
    0 references
    DLMF id
    DLMF:27.11.E12
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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.m2akdec
    0 references
    Möbius function
    Defining formula
    μ ⁡ ( n ) Moebius-mu 𝑛 {\displaystyle{\displaystyle\mu\left(\NVar{n}\right)}}
    xml-id
    C27.S2.E12.m2adec
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    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.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.m2aidec
    0 references
    Q12234
    Defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C27.S1.XMD4.m1gdec
    0 references
    Q12237
    Defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C27.S1.XMD6.m1kdec
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    Q12262
    Defining formula
    C 𝐶 {\displaystyle{\displaystyle C}}
    xml-id
    C27.S11.XMD2.m1dec
    0 references
     
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