Talk:Dirichlet's theorem on arithmetic progressions

Chowla proved this for the case of three consecutive terms.

What exactly did Chowla prove? AxelBoldt 02:20 Oct 10, 2002 (UTC)

Chowla proved there are infinitely many three consecutive terms in every arithmetic progression. We should also correct this piece of sentence:

...where n is a... --XJamRastafire 12:28 Oct 11, 2002 (UTC)

I think the part

Note that the theorem does not say that there are infinitely many consecutive terms in the arithmetic progression

a, a+d, a+2d, a+3d, ..., which are prime.

Would be easier to read if it said "Note that the theorem does not say that there are infinitely many consecutive prime terms in the arithmetic progression..."--or perhaps better: "...infinitely many consecutive primes...". —Preceding unsigned comment added by 74.185.249.234 (talk) 01:34, 5 March 2009 (UTC)

Euler's contribution

At present, the History section states: "Euler stated that every arithmetic progression beginning with 1 contains an infinite number of primes, … " This is incorrect.

I checked this claim about Euler, and Euler didn't say that. Euler's work on this problem appeared in: Leonard Euler (1737) "Variae observationes circa series infinitas" (Various observations about infinite series), Commentarii academiae scientiarum imperialis Petropolitanae, 9 : 160–188 ; specifically, Theorema 7 on pp. 172–174. Euler wasn't considering arithmetic progressions. Instead, he proved the sum-product formula for the Riemann zeta function for the value s=1. He did not conclude from that result that there were an infinitude of primes.

Euler's article is discussed in: C. Edward Sandifer, The Early Mathematics of Leonhard Euler (Providence, Rhode Island: The Mathematical Society of America, 2007), pp. 249–260. On p. 254, Sandifer states: "This result [i.e., Theorem 7] is sometimes cited as Euler's proof that there are infinitely many prime numbers. Because the harmonic series diverges, the product diverges, and a product can diverge only if it is an infinite product. Hence there must be infinitely many prime numbers. Though the infinitude of primes is immediate, Euler does not explicitly make the connection here."

VexorAbVikipædia (talk) 02:38, 19 July 2016 (UTC)

I don't know about Euler's contribution precisely, but currently the article states "Some preliminary work on the problem was done by the Swiss mathematician Leonhard Euler in 1737. Specifically, he proved the sum-product formula for the Riemann zeta function for the value s=1." I don't know what this is supposed to mean: the zeta function has a simple pole at s=1, is this what Euler proved? The main point in Dirichlet's proof is that the Dirichlet L-functions for nontrivial characters have neither pole nor zero at s=1 and this then gives rise to Dirichlet's theorem. VexorAbVikipædia, could you add a more precise statement what Euler actually proved and also look for a secondary source on the matter? Thanks! Jakob.scholbach (talk) 07:39, 21 July 2016 (UTC)

I have removed this unclear statement and provided a reference for the claim that the case a=1 can be proven in an elementary way. Jakob.scholbach (talk) 11:29, 9 August 2016 (UTC)

I apologize for the delay in replying to your comment of 21 July 2016. Wikipedia did not alert me about your comment.

Euler did not prove that the zeta function has a simple pole at s=1; instead, he showed that for s=1, the zeta function has a pole of at least order 1, but he did not prove that the order of the pole is exactly 1 (i.e., simple). Specifically, he proved that for the case of s=1, the zeta function reduces to a ratio of two infinite products, which appears in his paper of 1737 — which (incidentally) is mentioned later in the History section but which you did not delete, so the reader has no idea when Euler did work related to the zeta function or what that work was.

I'll replace "he proved the sum-product formula for the Riemann zeta function for the value s=1." with "he proved that for the value s=1, the Riemann zeta function reduces to a ratio of two infinite products, ∏ p / ∏ (p–1) , for all primes p."

Regarding your request for a secondary source, that's provided by (Sandifer, 2007), p. 253, which I cited and to which I provided a link.

VexorAbVikipædia (talk) 18:52, 9 August 2016 (UTC)

OK, that sounds more reasonable. However, this is, as far as I can see, unrelated to Dirichlet's theorem. (Euler's result implies that there are infinitely many primes, which is a much weaker statement than Dirichlet's theorem). For this reason, I would not include this statement here. Jakob.scholbach (talk) 08:36, 10 August 2016 (UTC)

Almost every account of Dirichlet's theorem on arithmetic progressions mentions both Euler's and Legendre's earlier work on the problem: Dirichlet's paper of 1837 cites Legendre's work, and Legendre, in turn, cites Euler's work. Since they cited their predecessors' work, that work should be included here, so that readers can conveniently discover and access the sources that Dirichlet and Legendre cite. (They apparently thought that it was worthy of note.) More importantly, some readers will wonder: where did Dirichlet and Legendre get the idea that a simple arithmetic progression might include an infinite number of primes? … and that the zeta function might be a route to a proof? Answer: Euler.

VexorAbVikipædia (talk) 16:39, 10 August 2016 (UTC)

I have tweaked this a bit: the fact that Euler did not at this point note the infinitude of primes is not terribly important, IMO. This is a basic fact known since Euclid. Also, formulas such as $\zeta (2)=\pi ^{2}/6$ (known to Euler) make it obvious that there are infinitely many primes. Let's not draw the reader's attention to minor bibliographical aspects of the history here. Jakob.scholbach (talk) 11:51, 15 August 2016 (UTC)