DLMF:27.5.E5 (Q8029)

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DLMF:27.5.E5
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    Statements

    Defining formula
    ln ⁡ n = ∑ d | n Λ ⁡ ( d ) ⟺ Λ ⁡ ( n ) = ∑ d | n ( ln ⁡ d ) ⁢ μ ⁡ ( n d ) . ⟺ 𝑛 subscript divides 𝑑 𝑛 Mangoldt-Lambda 𝑑 Mangoldt-Lambda 𝑛 subscript divides 𝑑 𝑛 𝑑 Moebius-mu 𝑛 𝑑 {\displaystyle{\displaystyle\ln n=\sum_{d\mathbin{|}n}\Lambda\left(d\right)% \Longleftrightarrow\Lambda\left(n\right)=\sum_{d\mathbin{|}n}(\ln d)\mu\left(% \frac{n}{d}\right).}}
    0 references
    DLMF id
    DLMF:27.5.E5
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    Symbols used
    Mangoldt’s function
    Defining formula
    Λ ⁡ ( n ) Mangoldt-Lambda 𝑛 {\displaystyle{\displaystyle\Lambda\left(\NVar{n}\right)}}
    xml-id
    C27.S2.E14.m2adec
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    Möbius function
    Defining formula
    μ ⁡ ( n ) Moebius-mu 𝑛 {\displaystyle{\displaystyle\mu\left(\NVar{n}\right)}}
    xml-id
    C27.S2.E12.m2acdec
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    principal branch of logarithm function
    Defining formula
    ln ⁡ z 𝑧 {\displaystyle{\displaystyle\ln\NVar{z}}}
    xml-id
    C4.S2.E2.m2adec
    0 references
    Q12241
    Defining formula
    d 𝑑 {\displaystyle{\displaystyle d}}
    xml-id
    C27.S1.XMD1.m1ddec
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