Riemann zeta function

Riemann zeta function

1

Definition

The

Riemann zeta function

is defined to be the

complex

valued

function

given by the series

ζ

⁢

(

s

)

:=

∑

n

=

1

∞

1

n

s

,

(1)

which is valid (in fact,

absolutely convergent

) for all

complex
numbers

s

with

Re

⁡

(

s

)

>

1

. We list here some of the key

properties

[

1

]

of the

zeta function

.

1.

For all

s

with

Re

⁡

(

s

)

>

1

, the zeta function

satisfies

the

Euler product formula

ζ

⁢

(

s

)

=

∏

p

1

1

-

p

-

s

,

(2)

where the

product

is taken over all

positive integer

primes

p

, and

converges
uniformly

in a neighborhood of

s

.

2.

The zeta function has a

meromorphic continuation

to the entire

complex plane

with a

simple pole

at

s

=

1

, of

residue

1

, and no
other singularities.

3.

The zeta function satisfies the

functional equation

ζ

⁢

(

s

)

=

2

s

⁢

π

s

-

1

⁢

sin

⁡

π

⁢

s

2

⁢

Γ

⁢

(

1

-

s

)

⁢

ζ

⁢

(

1

-

s

)

,

(3)

for any

s

∈

ℂ

(where

Γ

denotes the

Gamma function

).

2

Distribution of primes

The Euler product formula (

2

) given above expresses the
zeta function as a product over the primes

p

∈

ℤ

, and
consequently provides a link between the

analytic

properties of the
zeta function and the

distribution

of primes in the

integers

. As the
simplest possible illustration of this link, we show how the
properties of the zeta function given above can be used to prove that
there are infinitely many primes.

If the set

S

of primes in

ℤ

were finite, then the Euler product
formula

ζ

⁢

(

s

)

=

∏

p

∈

S

1

1

-

p

-

s

would be a finite product, and consequently

lim

s

→

1

⁡

ζ

⁢

(

s

)

would exist and would equal

lim

s

→

1

⁡

ζ

⁢

(

s

)

=

∏

p

∈

S

1

1

-

p

-

1

.

But the

existence

of this limit contradicts the fact that

ζ

⁢

(

s

)

has a pole at

s

=

1

, so the set

S

of primes cannot be finite.

A more sophisticated

analysis

of the zeta function along these lines
can be used to prove both the analytic

prime number theorem

and
Dirichlet’s

theorem

on primes in arithmetic progressions

1

1

In the case of

arithmetic progressions

, one also needs to examine the closely related Dirichlet

L

–functions in

addition

to the zeta function itself.

. Proofs of
the prime number theorem can be found in

[

2

]

and

[

5

]

, and for proofs of Dirichlet’s theorem on primes
in arithmetic progressions the reader may look in

[

3

]

and

[

7

]

.

3

Zeros of the zeta function

A

nontrivial zero

of the Riemann zeta function is defined to be
a root

ζ

⁢

(

s

)

=

0

of the zeta function with the property that

0

≤

Re

⁡

(

s

)

≤

1

. Any other zero is called

trivial

zero

of
the zeta function.

The reason behind the terminology is as follows. For complex numbers

s

with

real part

greater than 1, the series

definition

(

1

)
immediately shows that no zeros of the zeta function exist in this
region. It is then an easy matter to use the functional
equation (

3

) to find all zeros of the zeta function
with real part less than 0 (it turns out they are exactly the values

-

2

⁢

n

, for

n

a positive integer). However, for values of

s

with
real part between 0 and 1, the situation is quite different, since we
have neither a series definition nor a functional equation to fall
back upon; and indeed to this day very little is known about the
behavior of the zeta function inside this

critical strip

of the
complex plane.

It is known that the prime number theorem is

equivalent

to the
assertion that the zeta function has no zeros

s

with

Re

⁡

(

s

)

=

0

or

Re

⁡

(

s

)

=

1

. The celebrated

Riemann hypothesis

asserts that all nontrivial zeros

s

of the zeta function satisfy the much more precise equation

Re

⁡

(

s

)

=

1

/

2

. If true, the

hypothesis

would have profound

consequences

on the distribution of primes in the
integers

[

5

]

.

References

1

Lars Ahlfors

,

Complex Analysis

, Third Edition

,
McGraw–Hill, Inc., 1979.

2

Joseph Bak & Donald Newman,

Complex
Analysis, Second Edition

, Springer–Verlag, 1991.

3

Gerald Janusz,

Algebraic Number Fields

, Second
Edition

,

American Mathematical Society

, 1996.

4

Serge Lang,

Algebraic Number Theory

, Second
Edition

, Springer–Verlag, 1994.

5

Stephen Patterson,

Introduction

to the Theory
of the Riemann Zeta Function

, Cambridge University Press, 1988.

6

B.

Riemann

,

Ueber die Anzahl der Primzahlen unter einer gegebenen Grösse

,

http://www.maths.tcd.ie/pub/HistMath/People/Riemann/Zeta/

http://www.maths.tcd.ie/pub/HistMath/People/Riemann/Zeta/

7

Jean–Pierre Serre,

A Course in

Arithmetic

,
Springer–Verlag, 1973.

Title

Riemann zeta function

Canonical

name

RiemannZetaFunction

Date of creation

2013-03-22 12:38:01

Last modified on

2013-03-22 12:38:01

Owner

alozano (2414)

Last modified by

alozano (2414)

Numerical id

18

Author

alozano (2414)

Entry type

Definition

Classification

msc 11M06

Synonym

ζ

function

Related topic

AnalyticContinuationOfRiemannZeta

Related topic

DedekindZetaFunction

Related topic

DirichletSeries

Related topic

EulerProduct

Related topic

Complex

Related topic

EulerProductFormula2

Related topic

HarmonicSeriesOfPrimes

Defines

Euler product formula

Defines

Riemann hypothesis

Generated on Fri Feb 9 15:37:32 2018 by

LaTeXML