can we have a Taylor series expansion of the function $\frac{1}{\zeta(s)}$at $s=1$? if so how can we determine the radius of convergence of this expansion without assuming the truth of the riemann hypothesis?
2026-03-26 04:36:37.1774499797
can we have a Taylor series expansion of the function $\frac{1}{\zeta(s)}$?
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The zeros of $\zeta(s)$ are the poles of $1/\zeta(s)$. From the functional equation, we know $s=-2$ is a trivial zero and from computer calculations $\zeta(s)$ has no nontrivial zeros in $|\Im(s)|\le10$, so we conclude that the radius of convergence for the Taylor expansion of $1/\zeta(s)$ is $|(-2)-1|=3$.