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I'm trying to prove the following theorem.

Theorem (Zero free region):

There exists $C>0$ such that $\sigma > 1-\frac{C}{log(\vert t \vert +4)}\Rightarrow \zeta(\sigma + i t)\neq 0$.


In the proof I have a problem with one of the arguments, namely:

  • If $0<\delta\leq 2$ then $\frac{-\zeta'}{\zeta}(1+\delta) = \frac{1}{1+\delta -1} + O(1)$.

My question is simply why? I think it has something to do with the fact that $\zeta(s)$ has a simple pole for $s = 1$ but is otherwise analytic in the half-plane $\sigma >0$ (when we denote $s = \sigma + it)$. Someone who can clarify this?

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