ca.analysis and odes - Truncated product of RiemannZeta(1)?

Hi, 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 RiemannZeta function at s = +1. It diverges, but the expression for the function is RiemannZeta(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 RiemannZeta(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