In proving the weak zero free region of the zeta function given by (Hadamard, de la Vallee Poussin 1896) one has for $ \sigma' > 1$ and any $t' \in \mathbb{R}$ we have:

$$|\zeta(\sigma' + it')| = exp(Relog \zeta(\sigma' + it')) $$ $$ = exp(-Re\sum_{p}log(1- \frac{1}{p^{\sigma' + it'}})) $$ $$ = exp(Re \sum_{p} \sum_{k} \frac{1}{kp^{k(\sigma' + it')}})$$ $$ = exp(Re\sum_{p}\sum_{k}\frac{cos(t'klogp)}{kp^{k \sigma'}})$$

How is the last equality obtained? I tried using taylor series but couldn't get the result.



Write $$\frac{1}{p^{k(\sigma' + it')}} = p^{-k(\sigma' + it')} = p^{-k\sigma'} p^{-k it'} = p^{-k\sigma'} (e^{\log(p)})^{-kit'} = p^{-k\sigma'} e^{-i(t' k \log(p))}.$$

Then take the real part, reminding yourself that the real part of $e^{i\theta} = \cos(\theta)$ and that $\cos$ is an even function.


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