PlanetMath: prime counting function

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prime counting function

(Definition)

The prime counting function is a non-multiplicative function for any positive real number , denoted as $\pi(x)$ and gives the number of primes not exceeding . It usually takes a positive integer for an argument. The first few values of $\pi(n)$ for $n = 1, 2, 3, \ldots$ are $0, 1, 2, 2, 3, 3, 4, 4, 4, 4, 5, 5, 6, 6, 6, 6, 7, 7, 8, 8 \ldots$ (OEIS A000720 ).

The asymptotic behavior of $\pi(x) \sim x/\ln x$ is given by the prime number theorem. This function is closely related with Chebyshev's functions $\vartheta(x)$ and $\psi(x)$ .

"prime counting function" is owned by XJamRastafire.

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See Also: logarithmic integral

Keywords:

number theory

Attachments:: proof that exceeds the product of the primes up to (Proof) by PrimeFan

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Cross-references: Chebyshev's functions, function, prime number theorem, argument, integer, primes, number, real number, positive, non-multiplicative function
There are 26 references to this entry.

This is version 10 of prime counting function, born on 2002-06-27, modified 2006-12-01.
Object id is 3138, canonical name is PrimeCountingFunction.
Accessed 3928 times total.

Classification:

AMS MSC:	11A41 (Number theory :: Elementary number theory :: Primes)
	11A25 (Number theory :: Elementary number theory :: Arithmetic functions; related numbers; inversion formulas)
	11N05 (Number theory :: Multiplicative number theory :: Distribution of primes)

Pending Errata and Addenda

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