Re: sums involving squares of zeta function at even values

From: Robin Chapman (rjc_at_ivorynospamtower.freeserve.co.uk)
Date: 09/29/04 

Date: Wed, 29 Sep 2004 15:05:58 +0100

François Charton wrote:

> Hi,
>
> When dealing with infinite sums of the form
>
> Sum { f(n) Zeta(2n) }
>
> one can use many identities or functional relations, like
> Pi x cot (Pi x) = Sum { Zeta(2n) x^2n },
> identities for sums like : sum {(zeta(2n)-1)/n } etc...
>
> Now, I have a problem which involves squares of the zeta function at even
> integer values, ie sums which look like
>
> Sum { f(n) Zeta(2n)^2 }
>
> what is the way to deal with such sums?

For s > 1,
zeta(s)^2 = sum_{k=1}^infinity d(k)/k^s
where d(k) is the number of divisors of k.
This may or may not help in your problem .... 

-- 
Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html
"Lacan, Jacques, 79, 91-92; mistakes his penis for a square root, 88-9"
Francis Wheen, _How Mumbo-Jumbo Conquered the World_

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