DLMF:27.5.E7 (Q8031)

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DLMF:27.5.E7
No description defined

    Statements

    Defining formula
    G ⁢ ( x ) = ∑ m = 1 ∞ F ⁢ ( m ⁢ x ) m s ⟺ F ⁢ ( x ) = ∑ m = 1 ∞ μ ⁡ ( m ) ⁢ G ⁢ ( m ⁢ x ) m s , ⟺ 𝐺 𝑥 superscript subscript 𝑚 1 𝐹 𝑚 𝑥 superscript 𝑚 𝑠 𝐹 𝑥 superscript subscript 𝑚 1 Moebius-mu 𝑚 𝐺 𝑚 𝑥 superscript 𝑚 𝑠 {\displaystyle{\displaystyle G(x)=\sum_{m=1}^{\infty}\frac{F(mx)}{m^{s}}% \Longleftrightarrow F(x)=\sum_{m=1}^{\infty}\mu\left(m\right)\frac{G(mx)}{m^{s% }},}}
    0 references
    DLMF id
    DLMF:27.5.E7
    0 references
    Symbols used
    Möbius function
    Defining formula
    μ ⁡ ( n ) Moebius-mu 𝑛 {\displaystyle{\displaystyle\mu\left(\NVar{n}\right)}}
    xml-id
    C27.S2.E12.m2aedec
    0 references
    Q12239
    Defining formula
    m 𝑚 {\displaystyle{\displaystyle m}}
    xml-id
    C27.S1.XMD3.m1dec
    0 references
    Q12237
    Defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C27.S1.XMD6.m1adec
    0 references
    Q12250
    Defining formula
    F ⁢ ( s ) 𝐹 𝑠 {\displaystyle{\displaystyle F(s)}}
    xml-id
    C27.S5.XMD1.m1adec
    0 references
    Q12251
    Defining formula
    G ⁢ ( s ) 𝐺 𝑠 {\displaystyle{\displaystyle G(s)}}
    xml-id
    C27.S5.XMD2.m1adec
    0 references
     
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