DLMF:27.12.E3 (Q8087): Difference between revisions

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‎Changed an Item: Add constraint
‎Changed an Item: Add constraint
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.m2adec

Revision as of 14:27, 2 January 2020

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DLMF:27.12.E3
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    Statements

    Defining formula
    π ⁡ ( x ) = ⌊ x ⌋ - 1 - ∑ p j ≤ x ⌊ x p j ⌋ + ∑ r ≥ 2 ( - 1 ) r ⁢ ∑ p j 1 < p j 2 < ⋯ < p j r ≤ x ⌊ x p j 1 ⁢ p j 2 ⁢ ⋯ ⁢ p j r ⌋ , number-of-primes-less-pi 𝑥 𝑥 1 subscript subscript 𝑝 𝑗 𝑥 𝑥 subscript 𝑝 𝑗 subscript 𝑟 2 superscript 1 𝑟 subscript subscript 𝑝 subscript 𝑗 1 subscript 𝑝 subscript 𝑗 2 ⋯ subscript 𝑝 subscript 𝑗 𝑟 𝑥 𝑥 subscript 𝑝 subscript 𝑗 1 subscript 𝑝 subscript 𝑗 2 ⋯ subscript 𝑝 subscript 𝑗 𝑟 {\displaystyle{\displaystyle\pi\left(x\right)=\left\lfloor x\right\rfloor-1-% \sum_{p_{j}\leq\sqrt{x}}\left\lfloor\frac{x}{p_{j}}\right\rfloor+\sum_{r\geq 2% }(-1)^{r}\*\sum_{p_{j_{1}}<p_{j_{2}}<\cdots<p_{j_{r}}\leq\sqrt{x}}\left\lfloor% \frac{x}{p_{j_{1}}p_{j_{2}}\cdots p_{j_{r}}}\right\rfloor,}}
    0 references
    DLMF id
    DLMF:27.12.E3
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    constraint
    x ≥ 1 𝑥 1 {\displaystyle{\displaystyle x\geq 1}}
    0 references
    Symbols used
    Q11169
    Defining formula
    ⌊ x ⌋ 𝑥 {\displaystyle{\displaystyle\left\lfloor\NVar{x}\right\rfloor}}
    xml-id
    introduction.Sx4.p1.t1.r17.m4adec
    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.m2adec
    0 references
     
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