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DLMF:27.5.E6 - MaRDI portal

DLMF:27.5.E6 (Q8723)

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DLMF:27.5.E6
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    Statements

    DLMF defining formula
    G ⁢ ( x ) = ∑ n ≤ x F ⁢ ( x n ) ⟺ F ⁢ ( x ) = ∑ n ≤ x μ ⁡ ( n ) ⁢ G ⁢ ( x n ) , ⟺ 𝐺 𝑥 subscript 𝑛 𝑥 𝐹 𝑥 𝑛 𝐹 𝑥 subscript 𝑛 𝑥 Moebius-mu 𝑛 𝐺 𝑥 𝑛 {\displaystyle{\displaystyle G(x)=\sum_{n\leq x}F\left(\frac{x}{n}\right)% \Longleftrightarrow F(x)=\sum_{n\leq x}\mu\left(n\right)G\left(\frac{x}{n}% \right),}}
    0 references
    Symbols used
    Möbius function
    DLMF defining formula
    μ ⁡ ( n ) Moebius-mu 𝑛 {\displaystyle{\displaystyle\mu\left(\NVar{n}\right)}}
    xml-id
    C27.S2.E12.m2addec
    0 references
    positive integer
    DLMF defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C27.S1.XMD4.m1edec
    0 references
    real number
    DLMF defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C27.S1.XMD6.m1dec
    0 references
    Dirichlet series
    DLMF defining formula
    F ⁢ ( s ) 𝐹 𝑠 {\displaystyle{\displaystyle F(s)}}
    xml-id
    C27.S5.XMD1.m1dec
    0 references
    Dirichlet series
    DLMF defining formula
    G ⁢ ( s ) 𝐺 𝑠 {\displaystyle{\displaystyle G(s)}}
    xml-id
    C27.S5.XMD2.m1dec
    0 references

    Identifiers

     
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