I have two Dirichlet's serieses (convergence) that equal on a domain of the complex plane, I want to prove that their coefficients equal, I thought to do it by contradiction but I didn't succeed.
2026-03-28 21:50:51.1774734651
Dirichlet series of complex variable
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If $F(s),G(s)$ converge at $s_0$ then they converge and are analytic for $\Re(s) > \Re(s_0)$. If they are equal on some neigborhood of $s_0$ then by analytic continuation they are equal for $\Re(s) > \Re(s_0)$.
Let $a_n-b_n$ be the first non-zero coefficient of $F(s)-G(s)$. Then $$\lim_{s\to +\infty}n^s ( F(s)-G(s)) = a_n-b_n$$ which is a contradiction. Thus there is no such $n$ and $F(s),G(s)$ have the same coefficients.