[FOM] Some questions regarding irrational numbers

Timothy Y. Chow tchow at alum.mit.edu
Sun Mar 1 22:30:59 EST 2009

On Sun, 1 Mar 2009, I wrote:
> Well, then, the Turan hypergraph example might work for you

Another possible source of examples occurred to me: multiple zeta values.  
These are expressions of the form

zeta(s_1, ..., s_k)  :=  sum_{n_1 > ... > n_k > 0} prod_{j=1}^k (n_j)^(-s_j)

where the s_j are positive integers and s_1 > 1.  In the paper 
"Combinatorial aspects of multiple zeta values" by Borwein et al. 
(Electronic J. Combin. 5 (1998), R38), the authors formulate a
conjecture, and illustrate by giving a specific example:

 zeta(3,2,2,1,2) + zeta(2,2,3,2,1) + zeta(2,3,1,2,2)  =  (pi^10)/11!

I don't know if this particular identity has been proved since then, but 
if it has, I'm sure there are others like it that are still open, since 
multiple zeta values are still not fully understood.  To the question "Why 
don't we just compute it?" the answer is that "just computing" was how 
many of these identities were discovered in the first place.


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