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.

  • $\begingroup$ Almost everything is in wiki/prime_number_theorem and wiki/Riemann_hypothesis $\endgroup$ – reuns Oct 26 '16 at 1:43
  • 1
    $\begingroup$ And the analytic proof of the PNT is with $\zeta(s)$ has no zeros on $\sigma > 1- \frac{A}{\log |t|}$ for some $A$, and $\frac{1}{\zeta(s)} = \mathcal{O}(\log^7 t)$ and $\zeta(s) = \mathcal{O}(\log t),\zeta'(s) = \mathcal{O}(\log^2 t)$ on $\sigma > 1- \frac{A}{\log |t|}$ as $t \to \infty$. See page 66 of Titchmarsh's book for how to finish the proof from this $\endgroup$ – reuns Oct 26 '16 at 2:31

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