This is motivated by a recent  [question](https://mathoverflow.net/questions/26336/integer-valued-factorial-ratios) 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?



Chebyshev's approach to the distribution of primes
Erdős and Diamond proved in [**1**] that Chebyshev could have achieved sharper bounds for the asymptotic behavior of the prime counting function. Nevertheless, their proof does not shed any light on the first question that you posed because they took the PNT for granted throughout their note.


References:


[**1**] **H. G. Diamond; P. Erdős.** *On sharp elementary prime number estimates*, Enseign. Math. (2) 26 (1980) 313-321.


