Here are different links to Tychonoff's articles: Über die topologische Erweiterung von Räumen (springerlink.com/content/l656352441w67612/…). Ein Fixpunktsatz (springerlink.com/content/n61706447r886l58/…).

Thank you! Two more suggestions: (1) Replace "This one proves word for word as in he case" by "One proves this theorem word for word as in the case". (2) To type Cech with the caron, copy and paste "Čech" (without the quotation marks). (It should work.)

Very interesting!!! Thank you! --- I was wondering if you couldn't insert the contents of your comment into your answer, to make them more visible.

Thank you for your answer. I noticed that Walter Rudin gives this proof in his "Functional Analysis".

Dear KP Hart: Thank you for your answer. (It contains at least one typo ("the the proof").) --- Folland (Real Analysis) claims that the general statement is due to Cech (On bicompact spaces, Ann. of Math. 38 (1937), 823-844. MR 1503374). What's your opinion?

Dear Pietro, Pete, Spencer: In "L'intégration dans les groupes topologiques" Weil explains that his proof of the existence of a Haar measure on locally compact groups was obtained by hiding filters. I think the situations are very similar. (And I find Weil's proof incredibly beautiful - for the ideas and the style.)

(Cont.) I know that Hecke was a much greater mathematician than I, but if had to teach this topic, I would still give an elementary proof.

Dear Spencer and Pietro Majer: I have no argument against your point, but I still prefer Loomis's phrasing. (I'd lie if I said the contrary.) I often find myself in this kind of situation. The (unrelated) example that strikes me most is Hecke's "Lectures on the Theory of Algebraic Numbers", where he says: There are elementary proofs of the quadratic reciprocity, but "they possess rather the character of supplementary verification". So "we will dispense entirely with a presentation of an elementary proof". (Cont.)

Dear John: In Bourbaki's theory the "axiom of choice" is built-in. If you remove it, everything falls down. (You even loose the definition of the quantifiers.) The fact that a product of nonempty sets is nonempty is obvious (in this theory), and "Zorn's Lemma" is a theorem. (It would be interesting to know if people like Serre and Grothendieck have ever used the expression "axiom of choice" (in their writings).)

Dear Pietro Majer: I think Loomis's proof (the one I gave) is the same as the one you mention (due to H. Cartan if I'm not mistaken). Loomis hides the (ultra)filters. But I think he hides them very nicely. (Loomis credit the proof he gives to Bourbaki.)

Dear Amadeus: I had the impression that Munkres's proof was almost the same as the Loomis's (the one I gave), which was written long before. (Thank you for correcting me if I'm wrong.)

Thank you! Unfortunately I don't have access to JSTOR. Chernoff's proof is also in Folland's Real Analysis. Does anybody know a public link to Chernoff's paper?

Dear John Stillwell: As a bourbakist, the expression "axiom of choice" makes no sense to me. But I thank you very much for your comment. (Unrelated aside: Thank you very much also for your wonderful translation of Dirichlet! It changed my life!) Thank you to all contributors!

@PLC: I've started to read your notes - an activity I'd never have expected be so addictive! If I understand correctly your point of view on the PNT, the Selberg-Erdös approach is more your cup of tea than Newman's argument. I feel myself more attracted to the latter, but it's probably by laziness, and I don't doubt that Selberg and Erdös were great mathematicians. I also like your "softened" version of Serre's statement, exploiting the already known divergence of the series of the 1/p.

Thank you also for your notes! I'll take a close look at them. I take back the 10% and agree with your more cautious phrasing (math.uga.edu/~pete/4400DT.pdf, footnote p. 1): "In fact, with relatively little additional work, one can show that the primes are, in a certain precise sense, equidistributed among the $\varphi(N)$ possible congruence classes." (I suppose it's a - perhaps indirect - reference to Newman's argument.)

I tried to make a careful comparison between Serre, Soprunov (math.umass.edu/~isoprou/pdf/primes.pdf), and Kedlaya (www-math.mit.edu/~kedlaya/18.785/dirichlet.pdf). It seemed to me that, if you want to prove the PNT for AP (as opposed to the AP Theorem alone) with the same amount of details as, say, Serre, you need to expand the text by about 10%.