I'm reading the book Elementary Proof of Prime Number Theorem and it gives several equivalence of PRT, namely,
Three with similar expressions:
(A1) $\lim_{x\rightarrow \infty} \frac{\pi (x)\ln x}{x} =1$;
(A2) $\lim_{x\rightarrow \infty} \frac{\theta (x)}{x}=1$;
(A3) $\lim_{x\rightarrow \infty} \frac{\psi (x)}{x}=1$.
and three non-trivial ones with slightly different form:
(B1) $M(x):=\sum_{n\le x}\mu (n)=o(x)$;
(B2) $\sum_{n\le x}\frac{\mu (n)}{n}=o(1)$;
(B3) $\sum_{n\le x}\frac{\Lambda (n)}{n}=\ln x-\gamma +o(1)$;
(B4) $L(x):=\sum_{n\le x}\lambda (n)=o(x)$;
(B5) $\int_1^\infty \frac{\psi (t)-t}{t^2}=-\gamma -1$.
By now, I can prove the equivalence of (A1), (A2), (A3) by Chebyshev inequality, and the proof of (B1), (B2), (B3) also gave in the previous book. I also proved (B4) $\Leftrightarrow$ (B1) using the so-called hyperbolic summation method. I havn't proved (B5) yet and posted it here.
Now I want to collect a big list about PNT's forms. Could anybody suggest some other non-trivial form of PNT? The statement as well as proof will be greatly appreciated. Thank you.