I know, there was a similar question like this: Can the coefficients of a Dirichlet series be recovered? But i can see that in case when given function(series) has only positive real numbers as a domain, then Perron's formula is useless here. Could someone show me a other way to solve this problem? Thanks in advance.
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There are two ways to find the coefficients : either look at $\lim_{s \to \infty} F(s)$ to find $f(1)$ then at $\lim_{s \to \infty} n^s (F(s)-\sum_{m=1}^n f(m) m^{-s})$ to find $f(n)$, or note that $f(n)n^{-\sigma} = \lim_{T \to \infty} \frac{1}{2T} \int_{-T}^T F(\sigma+2i\pi t) n^{it} dt$ for $\sigma$ larger than the abscissa of absolute convergence. – reuns Dec 25 '18 at 13:58
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But i don't understand the second way. I wrote that given function is determined only for pure real (and positive) numbers. – mkultra Dec 25 '18 at 17:13
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By Perron's formula (see Apostol's book, Chapter 11), we have that $$f(x) = \lim_{T \rightarrow \infty} \int_{-T}^T F(\sigma+\imath t) x^{\sigma+\imath t} dt,$$ for $\sigma > \sigma_F$ such that the Dirichlet series $F$ is absolutely convergent. – mds Jun 02 '19 at 10:01