DLMF:27.12.E8 (Q8092): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: Euler’s totient / rank
 
Normal rank
Property / Symbols used: Euler’s totient / qualifier
 
Defining formula:

ϕ ⁡ ( n ) Euler-totient-phi 𝑛 {\displaystyle{\displaystyle\phi\left(\NVar{n}\right)}}

\Eulertotientphi[]@{\NVar{n}}
Property / Symbols used: Euler’s totient / qualifier
 
xml-id: C27.S2.E7.m2adec
Property / Symbols used
 
Property / Symbols used: exponential function / rank
 
Normal rank
Property / Symbols used: exponential function / qualifier
 
Defining formula:

exp ⁡ z 𝑧 {\displaystyle{\displaystyle\exp\NVar{z}}}

\exp@@{\NVar{z}}
Property / Symbols used: exponential function / qualifier
 
xml-id: C4.S2.E19.m2abdec
Property / Symbols used
 
Property / Symbols used: logarithmic integral / rank
 
Normal rank
Property / Symbols used: logarithmic integral / qualifier
 
Defining formula:

li ⁡ ( x ) logarithmic-integral 𝑥 {\displaystyle{\displaystyle\mathrm{li}\left(\NVar{x}\right)}}

\logint@{\NVar{x}}
Property / Symbols used: logarithmic integral / qualifier
 
xml-id: C6.S2.E8.m2acdec
Property / Symbols used
 
Property / Symbols used: principal branch of logarithm function / rank
 
Normal rank
Property / Symbols used: principal branch of logarithm function / qualifier
 
Defining formula:

ln ⁡ z 𝑧 {\displaystyle{\displaystyle\ln\NVar{z}}}

\ln@@{\NVar{z}}
Property / Symbols used: principal branch of logarithm function / qualifier
 
xml-id: C4.S2.E2.m2afdec
Property / Symbols used
 
Property / Symbols used: Q12239 / rank
 
Normal rank
Property / Symbols used: Q12239 / qualifier
 
Defining formula:

m 𝑚 {\displaystyle{\displaystyle m}}

m
Property / Symbols used: Q12239 / qualifier
 
xml-id: C27.S1.XMD3.m1dec
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.m1edec

Latest revision as of 14:29, 2 January 2020

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DLMF:27.12.E8
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    Statements

    Defining formula
    li ⁡ ( x ) ϕ ⁡ ( m ) + O ⁡ ( x ⁢ exp ⁡ ( - λ ⁢ ( α ) ⁢ ( ln ⁡ x ) 1 / 2 ) ) , logarithmic-integral 𝑥 Euler-totient-phi 𝑚 Big-O 𝑥 𝜆 𝛼 superscript 𝑥 1 2 {\displaystyle{\displaystyle\frac{\mathrm{li}\left(x\right)}{\phi\left(m\right% )}+O\left(x\exp\left(-\lambda(\alpha)(\ln x)^{1/2}\right)\right),}}
    0 references
    DLMF id
    DLMF:27.12.E8
    0 references
    constraint
    m ≤ ( ln ⁡ x ) α 𝑚 superscript 𝑥 𝛼 {\displaystyle{\displaystyle m\leq(\ln x)^{\alpha}}}
    0 references
    m ≤ ( ln ⁡ x ) α 𝑚 superscript 𝑥 𝛼 {\displaystyle{\displaystyle m\leq(\ln x)^{\alpha}}}
    0 references
    α > 0 𝛼 0 {\displaystyle{\displaystyle\alpha>0}}
    0 references
    Symbols used
    order not exceeding
    Defining formula
    O ⁡ ( x ) Big-O 𝑥 {\displaystyle{\displaystyle O\left(\NVar{x}\right)}}
    xml-id
    C2.S1.E3.m2abdec
    0 references
    Euler’s totient
    Defining formula
    ϕ ⁡ ( n ) Euler-totient-phi 𝑛 {\displaystyle{\displaystyle\phi\left(\NVar{n}\right)}}
    xml-id
    C27.S2.E7.m2adec
    0 references
    exponential function
    Defining formula
    exp ⁡ z 𝑧 {\displaystyle{\displaystyle\exp\NVar{z}}}
    xml-id
    C4.S2.E19.m2abdec
    0 references
    logarithmic integral
    Defining formula
    li ⁡ ( x ) logarithmic-integral 𝑥 {\displaystyle{\displaystyle\mathrm{li}\left(\NVar{x}\right)}}
    xml-id
    C6.S2.E8.m2acdec
    0 references
    principal branch of logarithm function
    Defining formula
    ln ⁡ z 𝑧 {\displaystyle{\displaystyle\ln\NVar{z}}}
    xml-id
    C4.S2.E2.m2afdec
    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.m1edec
    0 references
     
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