"The PNT obtained by statistical methods" In a famous book "What is Mathematics" by  Richard Courant, Herbert Robbins  the authors presented not a rigorous proof,   but "an argument that at least makes plausible the truth of Gauss's famous law of the distribution of primes". The main point is to prove the existence of smooth density function $W(x)$ wich has the meaning to the derivative of prime numbers distribution function $A(x)$. The first edition of the book 1941, so what is the current point of view on this and why this idea can not be the basis for full proof? Why it's not fine for mathematicians?
 A: 
The main point is to prove the existence of smooth density function
  $W(x)$

This is the main point that is missing from the argument in Courant & Robbins.  What they demonstrate is that, assuming that $W(x)$ exists, then it should asymptotically have the size predicted by PNT.
A result of very similar strength was already known by the mid-1800s (so 50 years before PNT was first proved): Chebyshev famously proved that $\pi(x) \asymp x/\log x$ and that if $\lim_{x\to\infty} \pi(x) (\log x)/x$ exists then it must be equal to $1$.
But there are plenty of places in mathematics where we can easily prove "if the limit exists then it must be $C$".  This is not enough evidence to conclude that the limit is $C$ because the limit might genuinely not exist.  For a toy example, take the series
$$S = \sum_{n\ge 0} (-1)^n = 1 - 1 + 1 - 1 + 1 - 1 + 1 - 1 + 1 \cdots. $$
If the sum $S$ converges then it must be equal to
$$ (1-1) + (1-1) + (1-1) + \cdots = 0;$$
This is certainly not strong enough to conclude that $S$ converges and is equal to $0$; if it were, then by the same logic we must also conclude that
$$ S = 1 + (-1+1) + (-1+1) + (-1+1) + \cdots = 1,$$
but a series can't have two limits.  The truth is that $S$ does not exist (well, one could make the case that $S$ sort of behaves like $\tfrac12$, but "sort of behaves like" is not the same as "equals").
The fact that we can compute a value assuming it exists does not prove that it exists, no matter how elegant the computation was.  Likewise, Chebyshev's work was accomplished by groundbreaking elementary methods, and while very impressive, it falls far short of actually proving the Prime Number Theorem.  Nor did Chebyshev claim that it did: he correctly saw that his theorem could not rule out the possibility that $\pi(x)$ varies forever between, say, $0.9 x/\log x$ and $1.1 x/\log x$, without ever converging towards $x/\log x$.
