prime number theorem

Define π(x) as the number of primes less than or equal to x. The prime number theoremMathworldPlanetmath asserts that


as x, that is, π(x)/xlogx tends to 1 as x increases. Here logx is the natural logarithmMathworldPlanetmathPlanetmathPlanetmath.

There is a sharper statement that is also known as the prime number theorem:


where li is the logarithmic integralDlmfDlmfMathworldPlanetmath defined as

lix=2xdtlogt=xlogx+1!x(logx)2++(k-1)!x(logx)k+O{x(logx)k+1},for any fixed k

and R(x) is the error term whose behavior is still not fully known. From the work of Korobov and Vinogradov on zeroes of Riemann zeta-function it is known that


for every θ>35. The unproven Riemann hypothesisMathworldPlanetmath is equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath to the statement that R(x)=O(x1/2logx).

There exist a number of proofs of the prime number theorem. The original proofs by Hadamard [4] and de la Vallée Poussin[7] called on analysisMathworldPlanetmath of behavior of the Riemann zeta functionDlmfDlmfMathworld ζ(s) near the line s=1 to deduce the estimates for R(x). For a long time it was an open problem to find an elementary proof of the prime number theorem (“elementary” meaning “not involving complex analysis”). Finally Erdős and Selberg [3, 6] found such a proof. Nowadays there are some very short proofs of the prime number theorem (for example, see [5]).


  • 1 Tom M. Apostol. Introduction to Analytic Number TheoryMathworldPlanetmath. Narosa Publishing House, second edition, 1990. 0335.10001.
  • 2 Harold Davenport. Multiplicative Number Theory. Markham Pub. Co., 1967. 0159.06303.
  • 3 Paul Erdős. On a new method in elementary number theory. Proc. Nat. Acad. Sci. U.S.A., 35:374–384, 1949. 0034.31403.
  • 4 Jacques Hadamard. Sur la distribution des zéros de la fonction ζ(s) et ses conséquences arithmétiques. Bull. Soc. Math. France, 24:199–220. 27.0154.01.
  • 5 Donald J. Newman. Simple analytic proof of the prime number theorem. Amer. Math. Monthly, 87:693–696, 1980. online at http://www.jstor.orgJSTOR.
  • 6 Atle Selberg. An elementary proof of the prime number theorem. Ann. Math. (2), 50:305–311, 1949. 0036.30604.
  • 7 Charles de la Vallée Poussin. Recherces analytiques sur la théorie des nombres premiers. Ann. Soc. Sci. Bruxells, 1897.
Title prime number theorem
Canonical name PrimeNumberTheorem
Date of creation 2013-03-22 11:45:18
Last modified on 2013-03-22 11:45:18
Owner bbukh (348)
Last modified by bbukh (348)
Numerical id 21
Author bbukh (348)
Entry type Theorem
Classification msc 11A41
Classification msc 55U10
Classification msc 55U30
Classification msc 55T25
Classification msc 55M05
Classification msc 55U15
Classification msc 81T25
Classification msc 81-XX
Classification msc 20G42
Classification msc 81R50
Classification msc 17B37
Classification msc 81Q60
Classification msc 81V05
Classification msc 81T05
Classification msc 55R40
Defines pi(x)
Defines logarithmic integral