DLMF:27.11.E9 (Q8078): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: order not exceeding / rank
 
Normal rank
Property / Symbols used: order not exceeding / qualifier
 
Defining formula:

O ⁡ ( x ) Big-O 𝑥 {\displaystyle{\displaystyle O\left(\NVar{x}\right)}}

\bigO@{\NVar{x}}
Property / Symbols used: order not exceeding / qualifier
 
xml-id: C2.S1.E3.m2ahdec
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.m2abdec
Property / Symbols used
 
Property / Symbols used: Q12121 / rank
 
Normal rank
Property / Symbols used: Q12121 / qualifier
 
Defining formula:

≡ modular-equivalence {\displaystyle{\displaystyle\equiv}}

\equivmod
Property / Symbols used: Q12121 / qualifier
 
xml-id: introduction.Sx4.p2.t1.r10.m10adec
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: 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.m1dec
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.m1adec
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.m1hdec

Latest revision as of 14:26, 2 January 2020

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DLMF:27.11.E9
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    Statements

    Defining formula
    ∑ p ≤ x p ≡ h ( mod k ) 1 p = 1 ϕ ⁡ ( k ) ⁢ ln ⁡ ln ⁡ x + B + O ⁡ ( 1 ln ⁡ x ) , subscript 𝑝 𝑥 modular-equivalence 𝑝 annotated ℎ pmod 𝑘 1 𝑝 1 Euler-totient-phi 𝑘 𝑥 𝐵 Big-O 1 𝑥 {\displaystyle{\displaystyle\sum_{\begin{subarray}{c}p\leq x\\ p\equiv h\!\!\!\!\!\pmod{k}\end{subarray}}\frac{1}{p}=\frac{1}{\phi\left(k% \right)}\ln\ln x+B+O\left(\frac{1}{\ln x}\right),}}
    0 references
    DLMF id
    DLMF:27.11.E9
    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.m2ahdec
    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.m2abdec
    0 references
    Q12121
    Defining formula
    ≡ modular-equivalence {\displaystyle{\displaystyle\equiv}}
    xml-id
    introduction.Sx4.p2.t1.r10.m10adec
    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
    Q12238
    Defining formula
    k 𝑘 {\displaystyle{\displaystyle k}}
    xml-id
    C27.S1.XMD2.m1dec
    0 references
    Q12235
    Defining formula
    p , p 1 , … 𝑝 subscript 𝑝 1 … {\displaystyle{\displaystyle p,p_{1},\ldots}}
    xml-id
    C27.S1.XMD5.m1adec
    0 references
    Q12237
    Defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C27.S1.XMD6.m1hdec
    0 references
     
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