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# Generalization of Merten'sMertens' theorem

One classical Merten's Mertens' theorem tells us that $$\prod_{p \leq n} (1-\frac{1}{p})^{-1} = e^\gamma \log n + \mathcal{O}(1).$$ It is now very natural to ask, whether we have some good estimate to $$\prod_{p \leq n} (1-\frac{1}{p^s})^{-1}$$ for, let's say, $s > 1$ real. Of course the limit is $\zeta(s)$ for growing $n$, and I would like to have some portion estimate - or something similar - in the form $$\prod_{p \leq n} (1-\frac{1}{p^s})^{-1} = k_n \zeta(s)$$ where $k_n$ is quite well estimated with respect to $n$.

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# Generalization of Merten's theorem

One classical Merten's theorem tells us that $$\prod_{p \leq n} (1-\frac{1}{p})^{-1} = e^\gamma \log n + \mathcal{O}(1).$$ It is now very natural to ask, whether we have some good estimate to $$\prod_{p \leq n} (1-\frac{1}{p^s})^{-1}$$ for, let's say, $s > 1$ real. Of course the limit is $\zeta(s)$ for growing $n$, and I would like to have some portion estimate - or something similar - in the form $$\prod_{p \leq n} (1-\frac{1}{p^s})^{-1} = k_n \zeta(s)$$ where $k_n$ is quite well estimated with respect to $n$.