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In a paper of Riemann, he states to following formal identity.

If $f(s)=\sum\limits_{k=1}^{\infty}\frac{a_k}{k^s}$ and $F(x)=\sum\limits_{k=1}^{\infty}a_kx^k$ then $$\Gamma(s)f(s)=\int\limits_{0}^{\infty}x^{s-1}F(e^{-x})dx$$ Now I am not one to question the masters so I will take this as given. What I am interested in however is whether or not it is possible to obtain a formula for $F(x)$ in terms of $f(s)$ using the identity above. I had a little look on the Wikipedia page for integral transforms, but to be quite honest I didn't really understand how they work or whether it is possible to apply them to this problem. Any thoughts?

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The inverse Mellin transform will allow you to recover $g(t) = F(e^{-t})$ from $f(s)$; then you can recover $F(x)$ from $g(t)$, by writing $g(-2\pi i t)$ as a Fourier series in $x=e^{-2\pi i t}$.

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