As$\ n \to \infty$, how does a product over the primes less than$\ p_n$ equal the same product over the primes less than$\ n$?
Let$\ p_n$ be the$\ n$-th prime. Is$\ \lim_{n\to\infty} \log \log n \prod_{i=1}^{\lfloor \log n \rfloor} \frac{p_i-1}{p_i}>0$?