[seqfan] Re: Double zeta function derivative

[Maximilian Hasler]

According to

http://www.wolframalpha.com/input/?i=zeta(s%2C1)
the slope of s -> zeta(s,1) in s=0 is
-log(2*Pi)/2 = -0.9189385332046727417803297364

Maximilian




On Tue, May 31, 2011 at 6:41 AM, Richard Mathar
<mathar at strw.leidenuniv.nl> wrote:
>
> There is a numerical evaluation of a derivative of a double zeta function
> in http://oeis.org/A190643 which I do not understand:
>
> By definition
> zeta(s1,s2) = sum_{j1>j2>0} 1/(j1^s1*j2^s2).
> summing over positive integers j1 and j2. Taking s2=1 defines
> zeta(s1,1) = sum_{j1>j2>0) 1/(j1^s1*j2).
> The derivative with respect to the first variable is
> (d/ds) zeta(s,1) = -sum_{j1>j2>0) log(j1)/(j1^s1*j2).
> In the limit of s1->0 we get
> zeta'(0,1) = -sum_{j1>j2>0) log(j1)/j2.
> The following questions arise:
> (i) Why is this value positive as claimed given that j1 and j2 are both
> positive in the region of summation?
> (ii) How does the double sum converge given that neither sum_{j1} log(j1)
> nor sum_{j2}/j2 converge?
> Is this some sort of renormalization/complex continuation?
> Did I miss that summation and derivation do not commute somewhere?
>
> Richard Mathar
>
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