Talk:Prime Number Theorem

Could we "quarantine" this proof? It seems wrong in quite significant ways and no-one seems up to correcting it at the moment. I may take a class on analytic number theory next year (which would include a proof of PNT) and will be in a better position to start correcting it, but my feeling is that it'd require a huge amount of groundwork. The "book" by Newman is kind of startling, it's only 80 pages of large-print A5 and seems very very brief... It seems more like "notes in analytic number theory" rather than any sort of book. Caliburn (talk) 20:57, 18 June 2022 (UTC)

By the "book by Newman" do you mean by Ingham? Sorry to be late to the party. --prime mover (talk) 05:54, 24 August 2022 (UTC)

I am not sure if the proof is wrong because it is very hard to read. But I think it is worth to keep this proof too, since the approach seems unique.

However someone needs to improve it. For example, I cannot decrypt the following message:

The average of the first $N$ coefficients of $\dfrac {\zeta'} {\zeta}$ tend to $-1$ as $N$ goes to infinity.

--Usagiop (talk) 21:40, 18 June 2022 (UTC)

The author of a book is an esteemed mathematician so I don't doubt the validity, but the "sketchy" proof given in the book (of which it comprises only a page or two) has been interpolated with (as far as I can see) very wrong details. There was apparently an error in the first edition which was replicated here, but I think the general method is correct. I would assume that just means that $\dfrac {a_1 + a_2 + \ldots + a_N} N \to -1$ as $N \to \infty$, where $a_1, a_2, \ldots$ are the coefficients of the Taylor expansion of $\zeta'/\zeta$. Caliburn (talk) 22:01, 18 June 2022 (UTC)

OK, now I gradually understood, consulting other sources. It is about the coefficients of Dirichlet series of $\zeta'/\zeta$. It is shown that the coefficients are $\sequence {-\map\Lambda n}_n$ and the average claim is as you said but for $a_n=-\map\Lambda n$. But it is very hard to understand this from the texts here. The explanations must be improved. --Usagiop (talk) 22:19, 18 June 2022 (UTC)