DLMF:27.4.E11 (Q8022)

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DLMF:27.4.E11
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    Statements

    Defining formula
    ∑ n = 1 ∞ σ α ⁡ ( n ) ⁢ n - s = ζ ⁡ ( s ) ⁢ ζ ⁡ ( s - α ) , superscript subscript 𝑛 1 divisor-sigma 𝛼 𝑛 superscript 𝑛 𝑠 Riemann-zeta 𝑠 Riemann-zeta 𝑠 𝛼 {\displaystyle{\displaystyle\sum_{n=1}^{\infty}\sigma_{\alpha}\left(n\right)n^% {-s}=\zeta\left(s\right)\zeta\left(s-\alpha\right),}}
    0 references
    DLMF id
    DLMF:27.4.E11
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    constraint
    ℜ ⁡ s > max ⁡ ( 1 , 1 + ℜ ⁡ α ) 𝑠 1 1 𝛼 {\displaystyle{\displaystyle\Re s>\max(1,1+\Re\alpha)}}
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    Symbols used
    Riemann zeta function
    Defining formula
    ζ ⁡ ( s ) Riemann-zeta 𝑠 {\displaystyle{\displaystyle\zeta\left(\NVar{s}\right)}}
    xml-id
    C25.S2.E1.m2agdec
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    sum of powers of divisors of $$n$$
    Defining formula
    σ α ⁡ ( n ) divisor-sigma 𝛼 𝑛 {\displaystyle{\displaystyle\sigma_{\NVar{\alpha}}\left(\NVar{n}\right)}}
    xml-id
    C27.S2.E10.m2adec
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    Q10811
    Defining formula
    ℜ ⁡ absent {\displaystyle{\displaystyle\Re}}
    xml-id
    C1.S9.E2.m1agdec
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    Q12234
    Defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C27.S1.XMD4.m1jdec
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