About the identity sum(n=0,\infty, zeta(2n)/pi^(2n) = -1/(2 tan 1)


Hi,

I posted that night a message after having (at 3 o'clock in my bed)
noticed that sum(n=0,\infty, zeta(2n)/pi^(2n)) was a Hurwitz number
(a continued fraction with a reguler polynomial pattern repeating)
with a very small period ; it should be a very simple number, but
which one ? I jumped out of my bed and asked Plouffe's inverter,
which told me about -1/(2*sin(1)/cos(1)). Sorry for having posted
something like that. I sent a rectification to sci.math where I also
posted for comments, but I had to go to work, and I like rather
explaining in more details the things on sci.math.research

Sorry if the previous post is arrived. It is correct, but quite ugly.

Now, the idea of the proof. Yesterday evening I computed the continued
fraction of sum(n=0,\infty, zeta(2n) z^n ). You can see this continued
fraction on my Home Page (in the signature), but at that time I didn't
notice that z=1/Pi^2 would lead to very easy simplifications, allowing
to get the unitary simple continued fraction. Thus, both steps are
quite easy to prove. The first one do it by simple manipulations of
the series, while the second one uses usual manipulations on continued
fractions. If it is worth it, I can typeset it.

But I have no idea of how new is this identity. Can you help
me on that point ?

Regards,

--
Thomas Baruchel --- Home Page: http://baruchel.free.fr/~thomas/
write to baruchel at the host called bluebottle dot com
?crire ? baruchel chez l'h?te nomm? bluebottle point com
(you will be asked for a confirmation the first time you write)
.