DLMF:27.11.E11 (Q8080)

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DLMF:27.11.E11
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    Statements

    Defining formula
    ∑ p ≤ x p ≡ h ( mod k ) ln ⁡ p p = 1 ϕ ⁡ ( k ) ⁢ ln ⁡ x + O ⁡ ( 1 ) , subscript 𝑝 𝑥 modular-equivalence 𝑝 annotated ℎ 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 id
    DLMF:27.11.E11
    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.m2ajdec
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    Euler’s totient
    Defining formula
    ϕ ⁡ ( n ) Euler-totient-phi 𝑛 {\displaystyle{\displaystyle\phi\left(\NVar{n}\right)}}
    xml-id
    C27.S2.E7.m2acdec
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    Q12121
    Defining formula
    ≡ modular-equivalence {\displaystyle{\displaystyle\equiv}}
    xml-id
    introduction.Sx4.p2.t1.r10.m10aadec
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    principal branch of logarithm function
    Defining formula
    ln ⁡ z 𝑧 {\displaystyle{\displaystyle\ln\NVar{z}}}
    xml-id
    C4.S2.E2.m2ahdec
    0 references
    Q12238
    Defining formula
    k 𝑘 {\displaystyle{\displaystyle k}}
    xml-id
    C27.S1.XMD2.m1adec
    0 references
    Q12235
    Defining formula
    p , p 1 , … 𝑝 subscript 𝑝 1 … {\displaystyle{\displaystyle p,p_{1},\ldots}}
    xml-id
    C27.S1.XMD5.m1cdec
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    Q12237
    Defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C27.S1.XMD6.m1jdec
    0 references
     
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