Dirichlet series

Dirichlet series

Let

(

λ

n

)

n

≥

1

be an

increasing sequence

of
positive real numbers tending to

∞

.
A

Dirichlet series

with

exponents

(

λ

n

)

is
a series of the form

∑

n

a

n

⁢

e

-

λ

n

⁢

z

where

z

and all the

a

n

are

complex numbers

.

An

ordinary Dirichlet series

is one having

λ

n

=

log

⁡

n

for all

n

.
It is written

∑

a

n

n

z

.

The best-known examples are the

Riemann zeta function

(in which

a

n

is the constant (

http://planetmath.org/ConstantFunction

)

1

) and the more general

Dirichlet L-series

(in which the mapping

n

↦

a

n

is

multiplicative

and

periodic

).

When

λ

n

=

n

, the Dirichlet series is just a

power series

in the variable

e

-

z

.

The following are the basic

convergence

properties of Dirichlet series.
There is nothing profound about their proofs, which can be found
in

[

1

]

and in various other works on

complex analysis

and

analytic
number theory

.

Let

f

⁢

(

z

)

=

∑

n

a

n

⁢

e

-

λ

n

⁢

z

be a Dirichlet series.

1.

If

f

converges

at

z

=

z

0

, then

f

converges uniformly

in the region

ℜ

⁡

(

z

-

z

0

)

≥

0

-

α

≤

arg

⁡

(

z

-

z

0

)

≤

α

where

α

is any

real number

such that

0

<

α

<

π

/

2

.
(Such a region is known as a “Stoltz angle”.)

2.

Therefore, if

f

converges at

z

0

, its sum defines a

holomorphic
function

on the region

ℜ

⁡

(

z

)

>

ℜ

⁡

(

z

0

)

, and moreover

f

⁢

(

z

)

→

f

⁢

(

z

0

)

as

z

→

z

0

within any Stoltz angle.

3.

f

=

0

identically if and only if all the

coefficients

a

n

are zero.

So, if

f

converges somewhere but not everywhere in

ℂ

, then
the domain of its convergence is the region

ℜ

⁡

(

z

)

>

ρ

for
some real number

ρ

, which is called the

abscissa of convergence

of the Dirichlet series.
The abscissa of convergence of the series

f

⁢

(

z

)

=

∑

n

|

a

n

|

⁢

e

-

λ

n

⁢

z

, if it exists,
is called the

abscissa of absolute convergence

of

f

.

Now suppose that the coefficients

a

n

are all real and nonnegative.
If the series

f

converges for

ℜ

⁡

(

z

)

>

ρ

, and the resulting

function

admits an

analytic

extension

(

http://planetmath.org/AnalyticContinuation

) to a neighbourhood of

ρ

,
then the series

f

converges in a neighbourhood of

ρ

.
Consequently, the domain of convergence of

f

(unless it is the whole
of

ℂ

) is

bounded

by a singularity at a point on the

real axis

.

Finally, return to the general case of any complex numbers

(

a

n

)

, but
suppose

λ

n

=

log

⁡

n

, so

f

is an ordinary Dirichlet series

∑

a

n

n

z

.

1.

If the

sequence

(

a

n

)

is bounded, then

f

converges absolutely

in the
region

ℜ

⁡

(

z

)

>

1

.

2.

If the

partial sums

∑

n

=

k

l

a

n

are bounded, then

f

converges
(not necessarily absolutely) in the region

ℜ

⁡

(

z

)

>

0

.

References

1

Jean-Pierre Serre.

A Course in Arithmetic

, chapter VI.

Springer-Verlag, 1973.

http://www.emis.de/cgi-bin/zmen/ZMATH/en/quick.html?type=html&an=0256.12001

Zbl 0256.12001.

2

E. C. Titchmarsh.

The Theory of Functions

.

Oxford Univ. Press, second edition, 1958.

http://www.emis.de/cgi-bin/zmen/ZMATH/en/quick.html?type=html&an=0336.30001

Zbl 0336.30001.

Title

Dirichlet series

Canonical name

DirichletSeries

Date of creation

2013-03-22 13:59:22

Last modified on

2013-03-22 13:59:22

Owner

bbukh (348)

Last modified by

bbukh (348)

Numerical id

8

Author

bbukh (348)

Entry type

Definition

Classification

msc 30B50

Related topic

DirichletLFunction

Related topic

RiemannZetaFunction

Related topic

DirichletLSeries

Defines

ordinary Dirichlet series

Defines

Stoltz angle

Defines

abscissa of convergence

Defines

abscissa of absolute convergence

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