DLMF:27.11.E5 (Q8074)

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DLMF:27.11.E5
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    Statements

    Defining formula
    ∑ n ≤ x σ α ⁡ ( n ) = ζ ⁡ ( α + 1 ) α + 1 ⁢ x α + 1 + O ⁡ ( x β ) , subscript 𝑛 𝑥 divisor-sigma 𝛼 𝑛 Riemann-zeta 𝛼 1 𝛼 1 superscript 𝑥 𝛼 1 Big-O superscript 𝑥 𝛽 {\displaystyle{\displaystyle\sum_{n\leq x}\sigma_{\alpha}\left(n\right)=\frac{% \zeta\left(\alpha+1\right)}{\alpha+1}x^{\alpha+1}+O\left(x^{\beta}\right),}}
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    DLMF id
    DLMF:27.11.E5
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    constraint
    β = max ⁡ ( 1 , α ) 𝛽 1 𝛼 {\displaystyle{\displaystyle\beta=\max(1,\alpha)}}
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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.m2addec
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    Riemann zeta function
    Defining formula
    ζ ⁡ ( s ) Riemann-zeta 𝑠 {\displaystyle{\displaystyle\zeta\left(\NVar{s}\right)}}
    xml-id
    C25.S2.E1.m2adec
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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.m2aadec
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