PlanetMath: prime number theorem

(more info)

Math for the people, by the people.

Main Menu

sections Encyclopædia
Papers
Books
Expositions

meta Requests (204)
Orphanage
Unclass'd
Unproven (490)
Corrections (7)
Classification

talkback Polls
Forums
Feedback
Bug Reports

downloads Snapshots
PM Book

information News
Docs
Wiki
ChangeLog
TODO List
Legalese
About

Entry average rating: No information on entry rating

prime number theorem

(Theorem)

Define $\pi (x)$ as the number of primes less than or equal to . The prime number theorem asserts that

$\displaystyle \pi (x) \sim \frac{x}{\log x}$

as $x \rightarrow \infty$ , that is, $\pi(x) / \frac{x}{\log x}$ tends to 1 as

increases. Here ${\log x}$ is the natural logarithm.

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

$\displaystyle \pi(x)={\mathrm{li}}x+R(x),$

where ${\mathrm{li}}$ is the logarithmic integral defined as

$\displaystyle {\mathrm{li}}x=\int_2^x \frac{dt}{\log t}=\frac{x}{\log x}+\frac{... ...}+\dotsb+\frac{(k-1)!x}{(\log x)^{k}}+O\left\{\frac{x}{(\log x)^{k+1}}\right\},$ for any fixed

and

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

$\displaystyle R(x)=O\left\{x \exp(-c(\theta)(\log x)^\theta)\right\}$

for every $\theta>\tfrac{3}{5}$ . The unproven Riemann hypothesis is equivalent to the statement that $R(x)=O(x^{1/2}\log x)$ .

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 analysis of behavior of the Riemann zeta function $\zeta(s)$ near the line $\Re s=1$ to deduce the estimates for . 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]).

Bibliography

1: Tom M. Apostol.
Introduction to Analytic Number Theory.
Narosa Publishing House, second edition, 1990.
Zbl 0335.10001.
2: Harold Davenport.
Multiplicative Number Theory.
Markham Pub. Co., 1967.
Zbl 0159.06303.
3: Paul Erdős.
On a new method in elementary number theory.
Proc. Nat. Acad. Sci. U.S.A., 35:374-384, 1949.
Zbl 0034.31403.
4: Jacques Hadamard.
Sur la distribution des zéros de la fonction $\zeta(s)$ et ses conséquences arithmétiques.
Bull. Soc. Math. France, 24:199-220.
JFM 27.0154.01.
5: Donald J. Newman.
Simple analytic proof of the prime number theorem.
Amer. Math. Monthly, 87:693-696, 1980.
Available online at JSTOR.
6: Atle Selberg.
An elementary proof of the prime number theorem.
Ann. Math. (2), 50:305-311, 1949.
Zbl 0036.30604.
7: Charles de la Vallée Poussin.
Recherces analytiques sur la théorie des nombres premiers.
Ann. Soc. Sci. Bruxells, 1897.

"prime number theorem" is owned by bbukh. [ full author list (2) | owner history (1) ]

(view preamble)

Also defines:

pi(x), logarithmic integral

Keywords:

number theory

Attachments:: bounds on $\pi(n)$ (Theorem) by rm50
values of the prime counting function and estimates for selected inputs (Example) by PrimeFan

Log in to rate this entry.
(view current ratings)

Cross-references: open problem, estimates, line, near, Riemann zeta function, Hadamard, equivalent, Riemann hypothesis, natural logarithm, primes, number
There are 19 references to this entry.

This is version 16 of prime number theorem, born on 2001-10-15, modified 2006-09-05.
Object id is 199, canonical name is PrimeNumberTheorem.
Accessed 17658 times total.

Classification:

AMS MSC:

11A41 (Number theory :: Elementary number theory :: Primes)

Pending Errata and Addenda

None.[ View all 7 ]

Discussion

forum policy

No messages.

Interact