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11: 27.2 Functions

… ►Euclid’s Elements (Euclid (1908, Book IX, Proposition 20)) gives an elegant proof that there are infinitely many primes. …There is great interest in the function $\pi \left(x\right)$ that counts the number of primes not exceeding $x$. … ►

27.2.2 $$\pi \left(x\right)=\sum _{p\le x}1.$$

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Defines:

$\pi \left(x\right)$: number of primes not exceeding a number

Symbols:

$p,p_1,\mathrm{\dots }$: prime numbers and $x$: real number

Permalink:

http://dlmf.nist.gov/27.2.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §27.2(i), §27.2 and Ch.27

►Gauss and Legendre conjectured that $\pi \left(x\right)$ is asymptotic to $x/\ln x$ as $x\to \mathrm{\infty }$: … ►It is the special case $k=2$ of the function $d_k\left(n\right)$ that counts the number of ways of expressing $n$ as the product of $k$ factors, with the order of factors taken into account. …