There is an easy proof of the PNT, just in a few lines, in the book by Julian Havil, "Gamma", pages 201-202. Specifically, Von Mangoldt's formula, which is very easy to derive:
$$
\psi(x) = x - \ln(2\pi) - \frac{1}{2}\ln(1 - x^{-2}) - \sum\_{\zeta(\rho) = 0} \frac{x^{\rho}}{\rho},
$$
implies the PNT, since from this equation it follows that $\psi(x)/x \to 1$.


I am a Physicist and my Mathematician colleagues regard this proof as non-elementary. I've never seen easier proof of the PNT than this one. Any other proofs of the PNT without complex analysis are very lengthy, tedious and difficult to understand (and actually non-elementary, therefore). But why this a very simple proof is regarded as non-elementary? Unfortunately, I could not get a clear answer. Could you please clarify?


EDIT: Von Mangoldt's formula itself is not sufficient to immediately claim that $\psi(x)/x \to 1$. However, we can consider [Newman's proof](https://www.tandfonline.com/doi/pdf/10.1080/00029890.1997.11990704) of the PNT. This proof is much easier and shorter than any existing non-elementary proofs of the PNT. Although this proof involves very basics of the complex analysis, it is, nevertheless, regarded as a non-elementary proof. Therefore, the difference between elementary and non-elementary proofs of the PNT is the presence of the complex analysis. I hope, I am not mistaken.



What is the difference between elementary and non-elementary proofs of the Prime Number Theorem?
To complement Will Sawin's answer, in the specific context of the prime number theorem, there are historically well-established notions of "elementary" and "non-elementary" proofs, stemming from Hardy's 1921 lecture:



> 
> No elementary proof of the prime number theorem is known, and one may ask whether it is reasonable to expect one. Now we know that the theorem is roughly equivalent to a theorem about an analytic function, the theorem that Riemann’s zeta function has no roots on a certain line. A proof of such a theorem, not fundamentally dependent on the theory of functions, seems to me extraordinarily unlikely. It is rash to assert that a mathematical theorem cannot be proved in a particular way; but one thing seems quite clear. We have certain views about the logic of the theory; we think that some theorems, as we say ‘lie deep’ and others nearer to the surface. If anyone produces an elementary proof of the prime number theorem, he will show that these views are wrong, that the subject does not hang together in the way we have supposed, and that it is time for the books to be cast aside and for the theory to be rewritten.
> 
> 
> 


The history behind is that while up-to-multiplicative constant bounds for the prime counting function were obtained, by Chebyshev, by an ingenious but *elementary* counting arguments, all the proofs of PNT known in 1921 were a lot more involved and went via Riemann's zeta function and it analytic properties.


An elementary proof was found in 1948 by Selberg and Erdős:



> 
> In this paper will be given a new proof of the prime-number theorem, which is elementary in the sense that it uses practically no analysis, except the simplest properties of the logarithm.
> 
> 
> 


This became a big deal, in a large part, because it answered the above-quoted challenge of Hardy. The shortest proof nowadays is [Newman's proof,](https://www.tandfonline.com/doi/pdf/10.1080/00029890.1997.11990704?casa_token=nqD5CeXgqLwAAAAA:-4CZupo4hWsPL-lLYDcWd_OstRk04V2JZnUK2dpQ6cdbhUZynZ_TqZ4e6hnfCc_C_313YPuwoLB-yUc) although using complex analysis, it is much more "elementary" than the proofs known to Hardy; it is an interesting question how would Hardy judge it.


