Talk:Riemann zeta function

Added Intro and History

I removed the word "positive" from the index set for the Eulerian product, as the prime numbers page already defines prime numbers as being positive. I also changed the domain of validity of the series representation to all complex numbers with real part greater than 1. This will make less sense to a general audience, who probably won't understand complex exponents, but they also won't understand "analytic continuation", or "convergence" for that matter. As it seems standard to describe the domain of validity as the largest possible, I chose to change it. However, the definition for all real numbers > 1 does determine the analytic continuation uniquely, so the original description was not wrong.

Perhaps the functional equation, zeros, and special values should be lumped under a single heading, and then other headings could be created to describe things like various representations of the function and important results involving the function (like applications)? Barry R. Smith 21:29, 27 March 2008 (CDT)

Infinite Series vs. Dirichlet Series

I noticed that before the series definition of the zeta function, the descriptive term "infinite series" was changed to "Dirichlet series". I believe it should be changed back. Dirichlet series are encountered later in a mathematics education than the zeta function, and much later than the special case of $\zeta (2)$ whose value is often given in a first course on infinite series.

The reader this article is aimed at is one who has not already encountered the zeta function, or at least not in the detail described in the article. As such, we cannot expect them to jump to the page on Dirichlet series and be able to puzzle out what it says there. Better would be to link them to infinite series, in case, say, they just heard about some famous problem called the "Riemann Hypothesis" involving something called "zeta", but they haven't encountered infinite series before they reach the page.

Dirichlet series of course would be a reasonable link to include in the "Related Links" subpage, and I think it should be confined to that page. —The preceding unsigned comment was added by Barry R. Smith (talk • contribs) 02:38, 12 November 2008 (UTC)

Your comments on infinite series are fine, but why "confine" mention of Dirichlet series to a subpage? I strongly disagree with that. Zeta is indeed the first example and naturally leads into that subject (even if the article is puzzling). Richard Pinch 07:22, 12 November 2008 (UTC)

I didn't mean to imply we necessarily should confine mention, but rather leave it only as a mention and a link. The zeta function naturally leads into a host of other ideas, including zeta functions which have no series interpretation (hence are not Dirichlet series), such as general Artin L-functions. We would need to mention all other zeta and L-functions as well -- Ruelle, Ihara, Hasse-Weil, etc. I suppose it would be simplest just to say something like "a vast array of analogues of the Riemanna zeta function have appeared in other fields (links can be found on the "related pages" subpage), many of which can be described as types of Dirichlet series".

—The preceding unsigned comment was added by Barry R. Smith (talk • contribs) 14:41, 12 November 2008 (UTC)

Well, that may be true of some of the more general zeta functions (like Ihara), but this article is about Riemann's zeta, which is. It's also worth pointing out that some of the more general zeta functions (like the Dedekind zeta function of a field) are naturally expressed as a sum over ideals rather than integers, but are still expressible Dirichlet series. The book of Hardy and Riesz also discusses more general Dirichler series. Richard Pinch 19:15, 12 November 2008 (UTC)

It sounds like we are agreed that the term "infinite series" is more appropriate for the introduction of the definition of the zeta function, so I'm gonna go ahead and change it back. You or I or someone else can add something mentioning Dirichlet series further down the article.

—The preceding unsigned comment was added by Barry R. Smith (talk • contribs) 21:48, 13 November 2008 (UTC)

Sounds good to me. Richard Pinch 07:30, 14 November 2008 (UTC)