prime number theorem

prime number theorem

Define

π

⁢

(

x

)

as the number of primes less than or equal to

x

. The

prime number theorem

asserts that

π

⁢

(

x

)

∼

x

log

⁡

x

as

x

→

∞

, that is,

π

⁢

(

x

)

/

x

log

⁡

x

tends to 1 as

x

increases. Here

log

⁡

x

is the

natural logarithm

.

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

π

⁢

(

x

)

=

li

⁡

x

+

R

⁢

(

x

)

,

where

li

is the

logarithmic integral

defined as

li

⁡

x

=

∫

2

x

d

⁢

t

log

⁡

t

=

x

log

⁡

x

+

1

!

⁢

x

(

log

⁡

x

)

2

+

⋯

+

(

k

-

1

)

!

⁢

x

(

log

⁡

x

)

k

+

O

⁢

{

x

(

log

⁡

x

)

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

R

⁢

(

x

)

=

O

⁢

{

x

⁢

exp

⁡

(

-

c

⁢

(

θ

)

⁢

(

log

⁡

x

)

θ

)

}

for every

θ

>

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

ζ

⁢

(

s

)

near the line

ℜ

⁡

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

]

).

References

1

Tom M. Apostol.

Introduction

to

Analytic Number Theory

.

Narosa Publishing House, second edition, 1990.

http://www.emis.de/cgi-bin/zmen/ZMATH/en/quick.html?type=html&an=0335.10001

Zbl
0335.10001.

2

Harold Davenport.

Multiplicative Number Theory

.

Markham Pub. Co., 1967.

http://www.emis.de/cgi-bin/zmen/ZMATH/en/quick.html?type=html&an=0159.06303

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.

http://www.emis.de/cgi-bin/zmen/ZMATH/en/quick.html?type=html&an=0034.31403

Zbl
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.

http://www.emis.de/cgi-bin/zmen/ZMATH/en/quick.html?type=html&an=27.0154.01

JFM 27.0154.01.

5

Donald J. Newman.

Simple analytic proof of the prime number theorem.

Amer. Math. Monthly

, 87:693–696, 1980.

http://links.jstor.org/sici?sici=0002-9890%28198011%2987%3A9%3C693%3ASAPOTP%3E2.0.CO%3B2-U

Available online
at

http://www.jstor.org

JSTOR.

6

Atle Selberg.

An elementary proof of the prime number theorem.

Ann. Math. (2)

, 50:305–311, 1949.

http://www.emis.de/cgi-bin/zmen/ZMATH/en/quick.html?type=html&an=0036.30604

Zbl
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

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msc 81R50

Classification

msc 17B37

Classification

msc 81Q60

Classification

msc 81V05

Classification

msc 81T05

Classification

msc 55R40

Defines

pi(x)

Defines

logarithmic integral

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