DLMF:27.4.E11 (Q8022): Difference between revisions

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Changed an Item: Add constraint
Changed an Item: Add constraint
 
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Property / Symbols used
 
Property / Symbols used: Riemann zeta function / rank
 
Normal rank
Property / Symbols used: Riemann zeta function / qualifier
 
Defining formula:

ζ ( s ) Riemann-zeta 𝑠 {\displaystyle{\displaystyle\zeta\left(\NVar{s}\right)}}

\Riemannzeta@{\NVar{s}}
Property / Symbols used: Riemann zeta function / qualifier
 
xml-id: C25.S2.E1.m2agdec
Property / Symbols used
 
Property / Symbols used: sum of powers of divisors of $$n$$ / rank
 
Normal rank
Property / Symbols used: sum of powers of divisors of $$n$$ / qualifier
 
Defining formula:

σ α ( n ) divisor-sigma 𝛼 𝑛 {\displaystyle{\displaystyle\sigma_{\NVar{\alpha}}\left(\NVar{n}\right)}}

\sumdivisors{\NVar{\alpha}}@{\NVar{n}}
Property / Symbols used: sum of powers of divisors of $$n$$ / qualifier
 
xml-id: C27.S2.E10.m2adec
Property / Symbols used
 
Property / Symbols used: Q10811 / rank
 
Normal rank
Property / Symbols used: Q10811 / qualifier
 
Defining formula:

absent {\displaystyle{\displaystyle\Re}}

\realpart@@
Property / Symbols used: Q10811 / qualifier
 
xml-id: C1.S9.E2.m1agdec
Property / Symbols used
 
Property / Symbols used: Q12234 / rank
 
Normal rank
Property / Symbols used: Q12234 / qualifier
 
Defining formula:

n 𝑛 {\displaystyle{\displaystyle n}}

n
Property / Symbols used: Q12234 / qualifier
 
xml-id: C27.S1.XMD4.m1jdec

Latest revision as of 14:17, 2 January 2020

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DLMF:27.4.E11
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    Statements

    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),}}
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    DLMF:27.4.E11
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    s > max ( 1 , 1 + α ) 𝑠 1 1 𝛼 {\displaystyle{\displaystyle\Re s>\max(1,1+\Re\alpha)}}
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    ζ ( s ) Riemann-zeta 𝑠 {\displaystyle{\displaystyle\zeta\left(\NVar{s}\right)}}
    C25.S2.E1.m2agdec
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    σ α ( n ) divisor-sigma 𝛼 𝑛 {\displaystyle{\displaystyle\sigma_{\NVar{\alpha}}\left(\NVar{n}\right)}}
    C27.S2.E10.m2adec
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    absent {\displaystyle{\displaystyle\Re}}
    C1.S9.E2.m1agdec
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    n 𝑛 {\displaystyle{\displaystyle n}}
    C27.S1.XMD4.m1jdec
    0 references