This is my first question. It appeared while solving a research problem in cryptography. I am computer science student, so I apologize for lack of mathematical rigor in this question. Thanks for any help.


Consider the Riemann zeta function at s = +1. It diverges, but the expression for the function is $\zeta(1) = \lim\_{n \rightarrow \infty} \sum\_{i = 1}^{n} \frac{1}{i}$ , the truncated sum of which are the $n$-th harmonic number, $\mathcal{H}(n)$. 


The question is, how about the expression $\zeta(1) = \lim\_{n \rightarrow \infty} \prod\_{\textrm{primes } p\_i \leq n} \frac{1}{1-p\_i^{-1}}$. is the value of the truncated product $\mathcal{H}(n)$ too? 


My simulations for large values of $n$ tells me that it is some function of $\log n$ (for example comparing the ratio of the function for $n$ and $n^2$ and $n^3$ etc) How do we prove this? 


In summary, what is the value of $\prod\_{\textrm{primes } p\_i \leq n} \frac{1}{1-p\_i^{-1}}$?
Thanks



Truncated product of $\zeta(1)$?
Formula (8) on [this page](http://mathworld.wolfram.com/MertensConstant.html) gives the result


$$\prod\_{p \le n} \frac1{1-p^{-1}} = e^\gamma \log n \,(1 + o(1)).$$


