DLMF:27.11.E11 (Q8080): 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: 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.m2ahdec
Property / Symbols used
 
Property / Symbols used: Q12238 / rank
 
Normal rank
Property / Symbols used: Q12238 / qualifier
 
Defining formula:

k 𝑘 {\displaystyle{\displaystyle k}}

k
Property / Symbols used: Q12238 / qualifier
 
xml-id: C27.S1.XMD2.m1adec
Property / Symbols used
 
Property / Symbols used: Q12235 / rank
 
Normal rank
Property / Symbols used: Q12235 / qualifier
 
Defining formula:

p , p 1 , 𝑝 subscript 𝑝 1 {\displaystyle{\displaystyle p,p_{1},\ldots}}

p,p_{1},\ldots
Property / Symbols used: Q12235 / qualifier
 
xml-id: C27.S1.XMD5.m1cdec
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.m1jdec

Latest revision as of 14:27, 2 January 2020

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DLMF:27.11.E11
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    Statements

    p x p h ( mod k ) ln p p = 1 ϕ ( k ) ln x + O ( 1 ) , subscript 𝑝 𝑥 modular-equivalence 𝑝 annotated pmod 𝑘 𝑝 𝑝 1 Euler-totient-phi 𝑘 𝑥 Big-O 1 {\displaystyle{\displaystyle\sum_{\begin{subarray}{c}p\leq x\\ p\equiv h\!\!\!\!\!\pmod{k}\end{subarray}}\frac{\ln p}{p}=\frac{1}{\phi\left(k% \right)}\ln x+O\left(1\right),}}
    0 references
    DLMF:27.11.E11
    0 references
    O ( x ) Big-O 𝑥 {\displaystyle{\displaystyle O\left(\NVar{x}\right)}}
    C2.S1.E3.m2ajdec
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    ϕ ( n ) Euler-totient-phi 𝑛 {\displaystyle{\displaystyle\phi\left(\NVar{n}\right)}}
    C27.S2.E7.m2acdec
    0 references
    modular-equivalence {\displaystyle{\displaystyle\equiv}}
    introduction.Sx4.p2.t1.r10.m10aadec
    0 references
    ln z 𝑧 {\displaystyle{\displaystyle\ln\NVar{z}}}
    C4.S2.E2.m2ahdec
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    k 𝑘 {\displaystyle{\displaystyle k}}
    C27.S1.XMD2.m1adec
    0 references
    p , p 1 , 𝑝 subscript 𝑝 1 {\displaystyle{\displaystyle p,p_{1},\ldots}}
    C27.S1.XMD5.m1cdec
    0 references
    x 𝑥 {\displaystyle{\displaystyle x}}
    C27.S1.XMD6.m1jdec
    0 references