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I don't think this is a valid proof, particularly the use of O notation at the end, or the series manipulations, From a non mathematical point of view I don't understand how a proof of the prime number theorem can be given in about a paragraph as is done here and yet others go at great length to find "elementary proofs", if I am wrong about this proof can someone correct me, I only ask because I thought 'proof wiki' was a some what reliable source. |
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There are many woeful technical errors, especially towards the end of this proof, but the overall strategy looks sound and can be repaired. Honestly, it feels like someone wrote down a sketch and then invited random first-year students to fill in the details with wishful thinking. I think it hardly speaks well for ProofWiki. One of the principal ingredients is $M(n) = o(n)$, which is itself equivalent to PNT. The proof of this that ProofWiki provides is egregious. It makes the baffling claim that $$\text{“Clearly}\displaystyle \sum_{n \le N} \frac{\mu(n)}{n} \ge \sum_{n\le N} \frac{\mu(n)}{N},”$$ which is not clear to any sane reader. It's actually false at around $N = 18500$ or so. Nevertheless, I expect one can deduce this by doing a legitimate partial summation from the convergence of $\sum \mu(n)/n$, which is proved on a separate page (this part does at least imitate the complex analysis content of Newman's proof, but I haven't looked at the details). With this powerful result in hand, it isn't that hard to obtain PNT by elementary methods (Dirichlet hyperbola). One should be careful to choose a cutoff adapted to the implied rate of decay of $o(n)$, but I think it does go through. It just happens that the ProofWiki argument relies on enough typographic miracles that it's "not even wrong": it strikes me as little better than the old saw $$\displaystyle\frac{\sin(x)}{n} = 6.$$ |
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I haven't checked the details in that page, but it says that the proof is a variant of Donald Newman's 4-page proof. It is short and elementary, yes.
See also Zagier's
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All the sections before "The Proof of the Prime Number Theorem" seem valid, they are very standard lemmas. But then they start using the following identity: $$\displaystyle \frac{1}{\zeta(z)} \left({ \zeta'(z) - \zeta^2 (z) }\right) = \left({ \sum_{n \mathop = 1}^\infty \frac{\mu(n)}{n^z} }\right) \left({ \left({ \sum_{n \mathop = 1}^\infty \frac{\log(n)}{n^z} }\right) - \left({ \sum_{n \mathop = 1}^\infty \frac{d(n)}{n^z} }\right) }\right)$$ which has a sign error! The derivative of the zeta functions should be negated: $$\displaystyle \frac{1}{\zeta(z)} \left({ \zeta'(z) - \zeta^2 (z) }\right) = \left({ \sum_{n \mathop = 1}^\infty \frac{\mu(n)}{n^z} }\right) \left({ -\left({ \sum_{n \mathop = 1}^\infty \frac{\log(n)}{n^z} }\right) - \left({ \sum_{n \mathop = 1}^\infty \frac{d(n)}{n^z} }\right) }\right)$$ They make use of it with the wrong sign afterwards. |
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