My bad.........i am so tired........it looks like they tried that........

3. History of the Prime Number Theorem

In 1798 Legendre published the first significant conjecture on the size of pi(x), when in his book Essai sur la Théie des Nombres he stated

Legendre:

pi(x) is approximately x/(log x - 1.08366)

Clearly Legendre's conjecture is equivalent to the prime number theorem, the constant 1.08366 was based on his limited table for values of pi(x) (which only went to x = 400,000).  In the long run 1 is a better choice than Legendre's 1.08366.
Gauss was also studying prime tables and came up with a different estimate (perhaps first considered in 1791), communicated in a letter to Encke in 1849 and first published in 1863.

Gauss:

pi(x) is approximately Li(x) (the principal value of integral of 1/log u from u=0 to u=x).

Notice again that Gauss' conjecture is equivalent to the prime number theorem.  Let's compare these estimates:

Table 3. Comparisons of approximations to pi(x)

x	pi(x)	Gauss' Li	Legendre	x/(log x - 1)	R(x)
1000	168	178	172	169	168.4
10000	1229	1246	1231	1218	1226.9
100000	9592	9630	9588	9512	9587.4
1000000	78498	78628	78534	78030	78527.4
10000000	664579	664918	665138	661459	664667.4
100000000	5761455	5762209	5769341	5740304	5761551.9
1000000000	50847534	50849235	50917519	50701542	50847455.4
10000000000	455052511	455055614	455743004	454011971	455050683.3

In this table Gauss' Li(x) is always larger than pi(x), this is true for all small x > 2.  However in 1914 Littlewood proved that pi(x)-Li(x) assumes both positive and negative values infinitely often.  In 1986 Te Riele showed there are more than 10¹⁸⁰ successive integers x for which pi(x)>Li(x) between 6.62^.10³⁷⁰ and 6.69^.10³⁷⁰.
Tchebycheff made the first real progress toward a proof of the prime number theorem in 1850, showing there exist positive constants a < 1 < b such that

a(x/log x) < pi(x) < b(x/log x)