[seqfan] Double zeta function derivative

Richard Mathar mathar at strw.leidenuniv.nl
Tue May 31 12:41:50 CEST 2011

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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