This is motivated by a recent question by Wadim.

The negative answer should be known, since t is very natural, in this case I would be happy to see any reference.

May Pafnuty Lvovich Chebyshev's approach to distribution of primes lead to PNT itself, if we replace $\frac{(30 n)! n!}{(15 n)! (10 n)! (6 n)!}$ to other integer ratios of factorials? If not, what are the best constants in asymptotic relation $$ c_1 \frac{n}{\log n}< \pi(n)< c_2 \frac{n}{\log n} $$ which may be obtained on this way?

transfinite diameterwell discussed in H. Montgomery's CBMS(?) lectures. It is known that this cannot give $c=1$ in the asymptotics $\pi(n)\sim cn/\log(n)$. – Wadim Zudilin May 29 '10 at 9:00