So the question that I'm working on is the following.
Show that $\Pi_{p\leq z}(1-\dfrac{1}{p})=\dfrac{C(1+\mathcal{o}(1))}{\log z}$.
First off I take logs and just work with the sum and thisis what I get. $$\sum_{p\leq z}\log (1-\dfrac{1}{p})=-\sum_{p\leq z}\sum_{n=1}^{\infty}\dfrac{1}{np^{n}} $$
now so now I want to show that this sum is $-\log\log z +C' +\mathcal{o}(\log x^{-1})$.
But I'm not sure on who to do this, anyone got any ideas?
$$\ln(n)=\sum_{d\mid n}\Lambda(d)$$ $$\ln(\lfloor x \rfloor!)=\sum_{n\leq x}\Lambda(n)\lfloor\frac{x}{n}\rfloor$$ $$x\ln(x)+O(x)=x\sum_{p\leq x}\frac{\ln(p)}{p}+O(x)$$ $$\sum_{p\leq x}\frac{\ln(p)}{p}=\ln(x)+O(1)$$ $$\text{ Now conclude your result using summation by parts}$$