prime number theorem


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prime number theorem

[¦prīm ¦nəm·bər ‚thir·əm]
(mathematics)
The theorem that the limit of the quantity [π(x)] (ln x)/ x as x approaches infinity is 1, where π(x) is the number of prime numbers not greater than x and ln x is the natural logarithm of x.

prime number theorem

(mathematics)
The number of prime numbers less than x is about x/log(x). Here "is about" means that the ratio of the two things tends to 1 as x tends to infinity. This was first conjectured by Gauss in the early 19th century, and was proved (independently) by Hadamard and de la Vall'ee Poussin in 1896. Their proofs relied on complex analysis, but Erd?s and Selberg later found an "elementary" proof.
References in periodicals archive ?
This qualitative feature differs fundamentally from the prime number theorem for ordinary prime numbers where all roots contribute to the remainder term.
Mathematicians have shown that if the hypothesis is true, it would bolster the prime number theorem, implying there are no wild statistical fluctuations in the distribution of primes.
There appears to be a similar relationship between this data and the prime number theorem of n/ln(n), but unlike the traditional prime number theorem where all primes are included, this data considers only valid Z primes (where the first shell is filled, [s.
This culminated in the proofs of the Prime Number Theorem by Hadamard and de la Vallee Poussin in 1896, via establishing that [zeta](s) does not have zeros on the line {1 + it, t E I[8}.
Today this result is known as The Prime Number Theorem.
The answer is given by the celebrated Prime Number Theorem, whose proof was one of the glories of late nineteenth-century mathematics.