DLMF:27.11.E5 (Q8074): Difference between revisions

From DRMF
Jump to navigation Jump to search
Changed an Item: Add constraint
Changed an Item: Add constraint
 
(3 intermediate revisions by the same user not shown)
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.m2adec
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.m2aadec
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.m1ddec
Property / Symbols used
 
Property / Symbols used: Q12237 / rank
 
Normal rank
Property / Symbols used: Q12237 / qualifier
 
Defining formula:

x 𝑥 {\displaystyle{\displaystyle x}}

x
Property / Symbols used: Q12237 / qualifier
 
xml-id: C27.S1.XMD6.m1ddec

Latest revision as of 14:25, 2 January 2020

No description defined
Language Label Description Also known as
English
DLMF:27.11.E5
No description defined

    Statements

    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),}}
    0 references
    DLMF:27.11.E5
    0 references
    β = max ( 1 , α ) 𝛽 1 𝛼 {\displaystyle{\displaystyle\beta=\max(1,\alpha)}}
    0 references
    O ( x ) Big-O 𝑥 {\displaystyle{\displaystyle O\left(\NVar{x}\right)}}
    C2.S1.E3.m2addec
    0 references
    ζ ( s ) Riemann-zeta 𝑠 {\displaystyle{\displaystyle\zeta\left(\NVar{s}\right)}}
    C25.S2.E1.m2adec
    0 references
    σ α ( n ) divisor-sigma 𝛼 𝑛 {\displaystyle{\displaystyle\sigma_{\NVar{\alpha}}\left(\NVar{n}\right)}}
    C27.S2.E10.m2aadec
    0 references
    n 𝑛 {\displaystyle{\displaystyle n}}
    C27.S1.XMD4.m1ddec
    0 references
    x 𝑥 {\displaystyle{\displaystyle x}}
    C27.S1.XMD6.m1ddec
    0 references