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In mathematics, the prime-counting function is the function counting the number of prime numbers less than or equal to some real number x. [1] [2] It is denoted by π(x) (unrelated to the number π). A symmetric variant seen sometimes is π 0 (x), which is equal to π(x) − 1 ⁄ 2 if x is exactly a prime number, and equal to π(x) otherwise.
Because the set of primes is a computably enumerable set, by Matiyasevich's theorem, it can be obtained from a system of Diophantine equations. Jones et al. (1976) found an explicit set of 14 Diophantine equations in 26 variables, such that a given number k + 2 is prime if and only if that system has a solution in nonnegative integers: [7]
The prime-counting function can be expressed by Riemann's explicit formula as a sum in which each term comes from one of the zeros of the zeta function; the main term of this sum is the logarithmic integral, and the remaining terms cause the sum to fluctuate above and below the main term. [99]
All prime numbers from 31 to 6,469,693,189 for free download. Lists of Primes at the Prime Pages. The Nth Prime Page Nth prime through n=10^12, pi(x) through x=3*10^13, Random primes in same range. Interface to a list of the first 98 million primes (primes less than 2,000,000,000) Weisstein, Eric W. "Prime Number Sequences". MathWorld.
An incremental formulation of the sieve [2] generates primes indefinitely (i.e., without an upper bound) by interleaving the generation of primes with the generation of their multiples (so that primes can be found in gaps between the multiples), where the multiples of each prime p are generated directly by counting up from the square of the ...
Meissel already found that for k ≥ 3, P k (x, a) = 0 if a = π(x 1/3).He used the resulting equation for calculations of π(x) for big values of x. [1]Meissel calculated π(x) for values of x up to 10 9, but he narrowly missed the correct result for the biggest value of x.
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