Let $\psi(x)=\sum\_{n\leq x} \Lambda(n)$, where $\Lambda(n)$ is the von Mangoldt function.
Then as Chebyshev showed, the following equality holds
$$\sum\_{n\leq x} \psi(x/n)=x\log(x)-x+O(\log(x)).$$
My question is, how far can one go towards proving the prime number theorem by only
using the above estimate and the fact that $\psi$ is increasing? Alternatively, is there a well-known example of an increasing function $\psi(x)$ for which the above equality holds, but $\lim\_{x\to \infty}\frac{\psi(x)}{x}\neq 1?$ Thank you very much.



A naive question about the prime number theorem
Theorem 1 in the following paper of Ingham shows that the stated estimate, together with $\psi$ being positive and nondecreasing, is 'enough' to deduce that $\psi(x)/x \to 1$:


A. E. Ingham: *Some Tauberian theorems connected with the prime number theorem*. J. London
Math. Soc. 20, 171–180 (1945). [Full text (paywalled)](https://londmathsoc.onlinelibrary.wiley.com/doi/abs/10.1112/jlms/s1-20.3.171)


In fact, this remains true even if one only has the weaker error term $o(x)$ in place of $O(\log{x})$. Of course, one has to be careful about what `enough' means here; Ingham's proof still uses the nonvanishing of $\zeta(s)$ on $\Re(s)=1$, which is the key input in the usual proofs of the prime number theorem.


