Hi, Mike. Using Google I found a paper at <http://www.cs.ust.hk/faculty/golin/pubs/MergeAv.ps> which proves how the Mellin-Perron transform can be used to get a recurrence fmla. for the Dirichlet coefficients from the function f(s) (assuming it converges for Re(s) > 2), which I think can be used to easily recover the coefficents under the convergence assumption. When you get to the paper, search for the text "recover the coefficients" -- or just go to page 6. Regards, Dan Asimov ----------------------------------------------------------------

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...How does one find the sequence a(n) for a Dirichlet generating function f(s)=sum(n=1 to inf) a(n)/n^s ?

This isn't directly the answer you seek, but I recalled Z. A. Melzak gives a transform to "Dirichletize" a Taylor series:

If f(x) has Taylor series coefficients An then the operator Ds, defined by

Ds f(x) = integral z^(s-1) f(x e^-z) dz / Gamma(s), z=0..inf

gives the function with Taylor coefficients An/n^s.

Then you let x=1 and you get the corresponding Dirichlet series function, call it g(s).

Now if you can manage to invert that operator you could presumably start with g(s), transform it into the corresponding Taylor series, and then pick the coefficients off f(x) with the usual derivative operators. The compositions of the inverse with those operators would give you the operators yielding the Dirichlet sequence.

Probably not much help, but maybe a start...

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-- Mike Stay staym@clear.net.nz http://www.cs.auckland.ac.nz/~msta039

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