DLMF:27.12.E6 (Q8090): Difference between revisions

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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.m2aadec
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.m2addec
Property / Symbols used
 
Property / Symbols used: number of primes not exceeding $$x$$ / rank
 
Normal rank
Property / Symbols used: number of primes not exceeding $$x$$ / qualifier
 
Defining formula:

π ⁡ ( x ) number-of-primes-less-pi 𝑥 {\displaystyle{\displaystyle\pi\left(\NVar{x}\right)}}

\nprimes@{\NVar{x}}
Property / Symbols used: number of primes not exceeding $$x$$ / qualifier
 
xml-id: C27.S2.E2.m2acdec
Property / Symbols used
 
Property / Symbols used: Q12241 / rank
 
Normal rank
Property / Symbols used: Q12241 / qualifier
 
Defining formula:

d 𝑑 {\displaystyle{\displaystyle d}}

d
Property / Symbols used: Q12241 / qualifier
 
xml-id: C27.S1.XMD1.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.m1cdec

Latest revision as of 14:28, 2 January 2020

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DLMF:27.12.E6
No description defined

    Statements

    Defining formula
    | π ⁡ ( x ) - li ⁡ ( x ) | = O ⁡ ( x ⁢ exp ⁡ ( - d ⁢ ( ln ⁡ x ) 3 / 5 ⁢ ( ln ⁡ ln ⁡ x ) - 1 / 5 ) ) . number-of-primes-less-pi 𝑥 logarithmic-integral 𝑥 Big-O 𝑥 𝑑 superscript 𝑥 3 5 superscript 𝑥 1 5 {\displaystyle{\displaystyle\left|\pi\left(x\right)-\mathrm{li}\left(x\right)% \right|=O\left(x\exp\left(-d(\ln x)^{3/5}\,(\ln\ln x)^{-1/5}\right)\right).}}
    0 references
    DLMF id
    DLMF:27.12.E6
    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.m2aadec
    0 references
    exponential function
    Defining formula
    exp ⁡ z 𝑧 {\displaystyle{\displaystyle\exp\NVar{z}}}
    xml-id
    C4.S2.E19.m2aadec
    0 references
    logarithmic integral
    Defining formula
    li ⁡ ( x ) logarithmic-integral 𝑥 {\displaystyle{\displaystyle\mathrm{li}\left(\NVar{x}\right)}}
    xml-id
    C6.S2.E8.m2aadec
    0 references
    principal branch of logarithm function
    Defining formula
    ln ⁡ z 𝑧 {\displaystyle{\displaystyle\ln\NVar{z}}}
    xml-id
    C4.S2.E2.m2addec
    0 references
    number of primes not exceeding $$x$$
    Defining formula
    π ⁡ ( x ) number-of-primes-less-pi 𝑥 {\displaystyle{\displaystyle\pi\left(\NVar{x}\right)}}
    xml-id
    C27.S2.E2.m2acdec
    0 references
    Q12241
    Defining formula
    d 𝑑 {\displaystyle{\displaystyle d}}
    xml-id
    C27.S1.XMD1.m1dec
    0 references
    Q12237
    Defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C27.S1.XMD6.m1cdec
    0 references
     
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