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 ?
Ingham, The Distribution of Prime Numbers, Cambridge University Press, Cambridge, 1990.
The distribution of prime numbers has been found to be useful in analyzing electron and nuclear energy levels.
We discuss a recent development connecting the asymptotic distribution of prime numbers with weighted potential theory.
distribution of prime numbers, polynomials, integer coefficients, weighted transfinite diameter, weighted capacity, potentials
For example, Rockmore relies on Sir Michael Berry's "music of the primes" analogy--Sir Michael actually played the "music" at a 1996 conference--to explain the relevance of the RH to the distribution of prime numbers.
Euler, Dirichlet and Rieman pioneered work leading to the study of the analytic properties of L-functions as they relate to the distribution of prime numbers.
A mathematical duo has made a surprising advance in understanding the distribution of prime numbers, those whole numbers divisible only by themselves and 1.
Huxley, The distribution of prime numbers, Oxford University Press, Oxford, 1972.
To gain insights into the somewhat irregular distribution of prime numbers, mathematicians have studied a variety of subsets of all primes.
The discovery of such a large Mersenne prime by a hit-or-miss approach actually has little value for computational number theorists interested in the distribution of prime numbers and related mathematical issues.

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