Re: [metatuning] Digest Number 980

Hi Gene,

> > But you have to prove that both converge
> > in the same places which is the tricky part
> > if you want to do it properly - that
> > the product of all those series
> > converges at the same points
> > as the original sum.
>
> That's straightforward undergraduate math when Re(s)>1 because of
> absolute convergence. That the product diverges when s=1 is Euler's
> proof of the infinitude of primes.

Rightio, by the integral test, in fact the proof is
given here:
http://www.math.hmc.edu/calculus/tutorials/convergence/

That leaves the case Re(s)<=1 which is the
most interesting one. There I was confused,
the original zeta function indeed diverges
everywhere for Re(s)<1 but you can make an analytic continuation
of it which though it doesn't satisfy the formula
elsewhere, is identical to it for Re(s)>1
and converges for Re(s)<1, and is unique
because of the very strong restrictions there are
on the possible form of analytic (infinitely differentiable)
functions when they are complex valued
over the complex plane. Which is what Reimann's
zeta functino actually is so it isn't
actually given by that formula for Re(s) = 0.5.
Only for Re(s) >1.

http://www.informationblast.com/Analytic_continuation.html

I'm not sure how much we covered as undergraduate
- it was long ago now.

I'm intersted to know about the tuning
connection if you feel like saying anything
about it.

Robert

--- In metatuning@yahoogroups.com, "Robert Walker" <robertwalker@n...>
wrote:

> I'm intersted to know about the tuning
> connection if you feel like saying anything
> about it.

I should put up a web page about it; I was going to do so after
writing a Wikipedia article on the Riemann-Siegel zeta function and
the Dirchlet eta function as references and never got around to it.