Deriving the Prime Number Theorem: Does this work? I have my own derivation which seems fairly simple, and I wonder if it is correct; and if it is, whether it has been previously written up. It’s based on the Sieve of Eratosthenes, and it goes like this. You know that when checking for primes, you only have to go up to the square root of the number you’re testing. So if you’re looking for primes in the vicinity of 1,000,000 you only have to work the sieve up to 1000.
Now, let’s say you’re checking for primes in the vicinity of 1,200,000. Now you have to work the sieve a little farther – in fact, up to about 1100. So the sieving process is going to be more thorough. That’s why the density of primes goes down as you move upwards.
How much more thorough is the sieve going to be? It will be more thorough by exactly the number of primes between 1000 and 1100. In short, the rate of change of the density function in the vicinity of 1000000 depends on the value of the density function in the vicinity of 1000. Can we construct a differential equation from these constraints?
Yes we can, and you will see that when we solve it we get 1/lnx, the desired result. At least that's what I get. I write up my solution in more detail in this blogpost: http://marty-green.blogspot.ca/2017/05/how-to-derive-prime-number-theorem.html
I wonder if anyone could verify if this is correct, or if it has been done already?
 A: It seems fair to answer the question in terms of the linked blog post and the older version of the idea of de Visme$^1$ from Mathematical Gazette noted in the comments.   
First, the expression $1/\log x$ in connection with the sequence of primes dates at least to 1793 (according to Gauss). So $1/\log x$ in no wise "eluded" mathematicians, but was known and all but proven when Hadamard and de la Vallee Poussin published their respective proofs in 1896.
De Visme (dV) in his article makes explicit some assumptions that are tacit in the blog post and the question here. The "density of primes" idea, as dV acknowledges, emerges from "[a]ssuming the divisibility of a number by various primes to be independent events..." And so the probability that a number is not divisible by 2,3,5,... is the product of probabilities $(1-1/2)(1-1/3)(1-1/5)...$    
There are already answers on this site which give a clear idea of the logical problems involved with treating sequence of primes in a probabilistic way [links to be inserted]. For one thing, it is hard to assign a meaning to the phrase "random prime" when the likelihood that a very large prime can be pulled from a hypothetical urn is nil. 
Also, dV seems to resort to the prime number theorem in his argument when he says, "...identifying $x^{1/2}$ with the largest prime less than $(x^2+h),$ a fair approximation...for $x$ large." Unless I misunderstand, this is just:  $ p(\pi(x))/ x \sim 1.$  
Because it doesn't seem necessary to the argument, I won't dwell on the second paragraph in the OP, which seems like a non-sequitur unless you make assumptions about the density of primes less than $\sqrt{x}.$ How do we know that is not rising, if not by the prime number theorem?
Someone wrote--I cannot recall where--that the difficulty of finding an elementary proof for the p.n.t. coupled with Dirichlet's success may account for the sense that complex numbers could give insight into the behavior of primes. At any rate, one is stuck with a trail of inference that leads to the peculiar assertion that proving the prime number theorem is equivalent to proving:

There are no roots $\rho$ of $\zeta(s)=0$ on the real line $\Re (s)=1.$ See Edwards, p. 68$^1$). 

The prime number theorem can be used without proof in most situations, so the bar for an heuristic derivation is high. On one hand the argument in the OP is simple and interesting, on the other it gives no insight into the real reason we know the theorem to be true. It cannot be called a derivation of the prime number theorem; but it might be called a mnemonic. It is a good exercise in the use of the sieve and formation of ODEs. Maybe it could be modified to give something stronger, or maybe de Visme's argument is the best one can do. 
$^1$ Hoffmman de Visme, The Density of Prime Numbers, Math. Gazette, vol. 45, no. 351 (1961) pp. 13-14.
$^2$ Intuitive or not, Chebyshev showed that $x/\log x\sim \pi(x)$ can be derived from $\psi(x)\sim x.$ This in turn can be derived from Mangoldt's forumla for $\psi(x),$
$$\psi(x)=x-\sum_\rho \frac{x^{\rho}}{\rho}+\sum_n \frac{x^{-2n}}{2n}+\text{const.} $$ Dividing both sides by $x$ we see that terms on the right must disappear if this is to work. See Edwards, Riemann's Zeta Function, pp. 68 ff.  
