Looking for the proof of an identity involving zeta function

[ziyuang]

This one: http://dlmf.nist.gov/25.8.E8.png
Can anyone offer some clues or references?
Thank you~

[ziyuang]

Trying Mobius inversion

[Valeri Astanoff]

Good day,

With a little help of by computer :

In[1]:= $Assumptions = Abs[z] < 1;

In[2]:= Sum[(Zeta[2*k]*z^(2*k))/k, {k, 1, Infinity}]

Out[2]= -EulerGamma + Log[Gamma[1 - z]] + Log[Gamma[1 + z]] -
EulerGamma Zeta[0, 1 - z] - EulerGamma Zeta[0, 1 + z]

In[3]:= % // FullSimplify

Out[3]= Log[Gamma[1 - z]] + Log[Gamma[1 + z]]

In[4]:= Gamma[1 - z]*Gamma[1 + z] == (Pi*z)/Sin[Pi*z] // FullSimplify

Out[4]= True

hth

--
Valeri

[Gerry]

Your equation seems to be related to :

Eq11(z) = sum_i=1^Infinity((1-zeta(2*i))*z^(2*i)/i)
Eq12(z) = -(log(1-z^2)+log(gamma(1-z))+log(gamma(1+z)));
Eq13(z) = -(log(1-z^2)+log(Pi*z*1/sin(Pi*z)));
Eq14(z) = -(Euler + log((z-1)*z*(z+1)));

Depending on the domain of z Eq11(z)=Eq1x(z)

[Rob Johnson]

In article <fb402d00-9400-4fd6...@o18g2000prh.googlegroups.com>,

This message uses UTF-8, so if your reader cannot read UTF-8,

ζ = zeta
π = pi
≠ = not equal

In ASCII, your equation is

oo
--- ζ(2k) 2k πz
> ----- z = ln( ------- ) [1]
--- k sin(πz)
k=1

So let's start

oo
--- ζ(2k) 2k
> ----- z
--- k
k=1

oo oo
--- --- 2k
= > > 1/k (z/n)
--- ---
k=1 n=1

oo
--- n n
= > ln( --- --- )
--- n-z n+z
n=1

oo
--- n
= > ln( --- ) [2]
--- n+z
n≠0

In <http://www.whim.org/nebula/math/infharmseries.html>, equation
[7] says

+oo
--- 1
> --- = π cot(πz) [3]
--- n+z
n=-oo

Integrating this, we get

--- n+z sin(πz)
ln(z) + > ln( --- ) = ln( ------- ) [4]
--- n π
n≠0

We get the constants of integration in [4] by looking near z = 0.

Negating [4] and moving ln(z) to the right hand side, we get

--- n πz
> ln( --- ) = ln( ------- ) [5]
--- n+z sin(πz)
n≠0

Combining [2] and [5], we get [1].

Rob Johnson <r...@trash.whim.org>
take out the trash before replying
to view any ASCII art, display article in a monospaced font

[Rob Johnson]

[article below posted to the newsgroups listed in the header]

In article <2011030...@whim.org>,

Typo. The "oo" above the summation should not be there. The line
should read

--- n
= > ln( --- ) [2]
--- n+z
n≠0

meaning the principal value of the sum over all non-zero n (positive
and negative).

[ziyuang]

On Mar 3, 1:36 am, Rob Johnson <r...@trash.whim.org> wrote:
> In article <fb402d00-9400-4fd6-96d2-4b049301b...@o18g2000prh.googlegroups.com>,

Wow, thank you so much Rob. I really appreciate it. It must be a hard
work to type the formulas.
And thank other guys~

I've just found a note about it.
http://scipp.ucsc.edu/~haber/ph116A/pibern_11.pdf