How to find the coefficent of a term in a Dirichlet generating function in Mathematica?

Question

For a normal Dirichlet generating function like $Zeta[s]^2$, I can get the coefficient of the n-th term by applying Dirichlet convolution of the two constant functions. But how to find the coefficient of the n-th term of a Dirichlet generating functions like $1/(2-Zeta[s])$ or $Zeta[s-1]$? It seems there is no function in Mathematica to find the coefficient directly(c.f. to get the coefficient of a formal power series by using SeriesCoefficient[...]). Please be advised, thanks!

Answer 1

Let $F(s)=\sum_na_nn^{-s}$ be some Dirichlet series that converges absolutely at $\Re(s)=\sigma$. Then we have

$$ \lim_{T\to\infty}{1\over2iT}\int_{\sigma-iT}^{\sigma+iT}F(s)x^s\mathrm ds= \begin{cases} a_n & x=n \\ 0 & \text{otherwise} \end{cases} $$

Perhaps plugging this integral into Mathematica would work.

Answer 2

I found a way to manually calculate one term's coefficient by substituting the term $(Zeta[s] - 1)$ to the original series and then expand it into a Maclaurin formal power series . For a given n, the coefficient can be found by ordered factorization enumeration.