nt.number theory - Chebyshev's approach to the distribution of primes

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?