zeros of Dirichlet eta function

zeros of Dirichlet eta function

As stated in the parent entry (

http://planetmath.org/AnalyticContinuationOfRiemannZetaToCriticalStrip

), the definition of the

Riemann zeta function

may be analytically continued (

http://planetmath.org/AnalyticContinuation

) from the half-plane

ℜ

⁡

s

>

1

to the half-plane

ℜ

⁡

s

>

0

by using the

Dirichlet eta function

η

⁢

(

s

)

via the equation

ζ

⁢

(

s

)

=

η

⁢

(

s

)

1

-

2

2

s

.

(1)

Then only the status of the points

s

n

:=

1

+

n

⋅

2

⁢

π

⁢

i

ln

⁡

2

(

n

∈

ℤ

)

(2)

which are the zeros of

1

-

2

2

s

, remains :  are they poles of

ζ

⁢

(

s

)

or not?  E. Landau has 1909 signaled this problem, which has been elementarily solved not earlier than after 40 years, by D. V. Widder.  He proved that those numbers, except

s

=

1

,  are also zeros of

η

⁢

(

s

)

.  This means that they only are

removable singularities

of

ζ

⁢

(

s

)

and that (1) in fact extends

ζ

⁢

(

s

)

to every points of the half-plane

ℜ

⁡

s

>

0

except

s

=

1

.

A new direct proof by J. Sondow of the vanishing of the Dirichlet eta function at the points

s

n

≠

1

was published in 2003.  It is based on a relation between the

partial sums

η

n

⁢

(

s

)

and

ζ

n

⁢

(

s

)

of the series defining respectively the

functions

η

⁢

(

s

)

and

ζ

⁢

(

s

)

for

ℜ

⁡

s

>

1

, which involves the approximation of an

integral

by a

Riemann sum

.

With some clever but not so complicated performed on finite sums, Sondow writes for any

s

the following:

η

2

⁢

n

⁢

(

s

)

=

1

-

1

2

s

+

1

3

s

-

1

4

s

+

-

…

+

(

-

1

)

2

⁢

n

-

1

(

2

⁢

n

)

s

=

1

+

1

2

s

+

1

3

s

+

1

4

s

+

…

+

(

-

1

)

2

⁢

n

-

1

(

2

⁢

n

)

s

-

2

⁢

(

1

2

s

+

1

4

s

+

…

+

1

(

2

⁢

n

)

s

)

=

(

1

-

2

2

s

)

⁢

ζ

2

⁢

n

⁢

(

s

)

+

2

2

s

⁢

(

1

(

n

+

1

)

s

+

…

+

1

(

2

⁢

n

)

s

)

=

(

1

-

2

2

s

)

⁢

ζ

2

⁢

n

⁢

(

s

)

+

2

⁢

n

(

2

⁢

n

)

s

⁢

1

n

⁢

(

1

(

1

+

1

/

n

)

s

+

…

+

1

(

1

+

n

/

n

)

s

)

Now if

t

is real,

s

=

1

+

i

⁢

t

,  and

2

1

-

s

=

2

-

i

⁢

t

=

1

,  then the factor multiplying

ζ

2

⁢

n

⁢

(

s

)

is zero and consequently

η

2

⁢

n

⁢

(

s

)

=

1

n

i

⁢

t

⁢

R

n

⁢

(

1

/

(

1

+

x

)

s

,

0

,

1

)

where

R

n

⁢

(

f

⁢

(

x

)

,

a

,

b

)

denotes a special Riemann sum approximating the integral of

f

⁢

(

x

)

over

[

a

,

b

]

.  For

s

=

1

,  i.e.

t

=

0

, one gets

η

⁢

(

1

)

=

lim

n

→

∞

⁡

η

2

⁢

n

⁢

(

1

)

=

lim

n

→

∞

⁡

R

n

⁢

(

1

/

(

1

+

x

)

,

0

,

1

)

=

∫

0

1

d

⁢

x

1

+

x

=

ln

⁡

2

and otherwise, when

t

≠

0

,  one has

|

n

1

-

s

|

=

|

n

-

i

⁢

t

|

=

1

,  giving

|

η

⁢

(

s

)

|

=

lim

n

→

∞

⁡

|

η

2

⁢

n

⁢

(

s

)

|

=

lim

n

→

∞

⁡

|

R

n

⁢

(

1

/

(

1

+

x

)

s

,

0

,

1

)

|

=

|

∫

0

1

d

⁢

x

(

1

+

x

)

s

|

=

|

2

1

-

s

-

1

1

-

s

|

=

|

1

-

1

-

i

⁢

t

|

=

0

.

Note.

By (1) the Dirichlet eta function has as zeros
also the zeros of the Riemann zeta function (see

Riemann hypothesis

(

http://planetmath.org/RiemannZetaFunction

)).

References

1

E. Landau

:

Handbuch der Lehre von der Verteilung der Primzahlen

. Erster Band. Berlin (1909); p. 161, 933.

2

D. V. Widder

:

The

Laplace transform

.  Princeton University Press (1946); p. 230.

3

J. Sondow

: “Zeros of the

alternating zeta function

on the line

ℜ

⁡

s

=

1

”.  —

Amer. Math. Monthly

110

(2003).  Also available

http://arxiv.org/abs/math.NT/0209393

here.

4

J. Sondow

: “The Riemann hypothesis, simple zeros, and the

asymptotic

convergence

degree of improper Riemann sums”.  —

Proc. Amer. Math. Soc.

126

(1998).  Also available

http://www.ams.org/journals/proc/1998-126-05/S0002-9939-98-04607-3/

here.

Title

zeros of Dirichlet eta function

Canonical name

ZerosOfDirichletEtaFunction

Date of creation

2014-11-21 21:17:02

Last modified on

2014-11-21 21:17:02

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

19

Author

pahio (2872)

Entry type

Derivation

Classification

msc 30D30

Classification

msc 30B40

Classification

msc 11M41

Related topic

DirichletEtaFunction

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