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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.
For example, π(10) = 4 because there are four prime numbers (2, 3, 5 and 7) less than or equal to 10. The prime number theorem then states that x / log x is a good approximation to π(x) (where log here means the natural logarithm), in the sense that the limit of the quotient of the two functions π(x) and x / log x as x increases without ...
Riemann's original use of the explicit formula was to give an exact formula for the number of primes less than a given number. To do this, take F(log(y)) to be y 1/2 /log(y) for 0 ≤ y ≤ x and 0 elsewhere. Then the main term of the sum on the right is the number of primes less than x.
where ⌊ x ⌋ is the floor function, which denotes the greatest integer less than or equal to x and the p i run over all primes ≤ √ x. [1] [2] Since the evaluation of this sum formula becomes more and more complex and confusing for large x, Meissel tried to simplify the counting of the numbers in the Sieve of Eratosthenes. He and Lehmer ...
However, it does not contain all the prime numbers, since the terms gcd(n + 1, a n) are always odd and so never equal to 2. 587 is the smallest prime (other than 2) not appearing in the first 10,000 outcomes that are different from 1. Nevertheless, in the same paper it was conjectured to contain all odd primes, even though it is rather inefficient.
There exists a natural number N such that every even integer n larger than N is a sum of a prime less than or equal to n 0.95 and a number with at most two prime factors. Tomohiro Yamada claimed a proof of the following explicit version of Chen's theorem in 2015: [ 7 ]
In other words: Ramanujan primes are the least integers R n for which there are at least n primes between x and x/2 for all x ≥ R n. The first five Ramanujan primes are thus 2, 11, 17, 29, and 41. Note that the integer R n is necessarily a prime number: () (/) and, hence, () must increase by obtaining another prime at x = R n. Since ...
The Chebyshev functions, especially the second one ψ (x), are often used in proofs related to prime numbers, because it is typically simpler to work with them than with the prime-counting function, π (x) (see the exact formula below.) Both Chebyshev functions are asymptotic to x, a statement equivalent to the prime number theorem.