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• CommentRowNumber1.
• CommentAuthorzskoda
• CommentTimeApr 24th 2013

• CommentRowNumber2.
• CommentAuthorUrs
• CommentTimeJul 23rd 2014

I have edited the Idea-section at zeta function to make it just a tad more substantial. It’s still a stub though. But I don’t have time for more now.

• CommentRowNumber3.
• CommentAuthorUrs
• CommentTimeJul 23rd 2014

and I have started to split off some entries

(all mostly stubs, still)

Then I arranged them all a bit better (I hope) in the function field analogy – table

• CommentRowNumber4.
• CommentAuthorDavid_Corfield
• CommentTimeJul 23rd 2014

Is that Weil zeta function to be what Wikipedia has as Hasse-Weil zeta function? What should go in the table of all those zeta functions? The arithmetic one?

• CommentRowNumber5.
• CommentAuthorUrs
• CommentTimeJul 23rd 2014

Hasse-Weil is for number fields,no? The entry means the version for varieties over $\mathbb{F}_p$ that enters the Weil conjectures.

This naming after people is really bad, as one sees here. Somebody should have stopped Weil from having more than one idea. But I can’t help it.

(And also my battery dies now any second…)

• CommentRowNumber6.
• CommentAuthorDavid_Corfield
• CommentTimeJul 23rd 2014

Ah yes.

There are now two rows of zeta functions on the table which are separated. Presumably they’d be better placed adjacent.

• CommentRowNumber7.
• CommentAuthorUrs
• CommentTimeJul 23rd 2014
• (edited Jul 23rd 2014)

Right, so the idea of the table is that in the first half is the case of the affine/projective line and in the second half the case of its ramified covering curves.

For instance previously I had the Riemann zeta function and the Dedekind zeta function in the same box. But really the Dedekind zeta function is the generalization of the Riemann zeta function from $Spec(\mathbb{Z})$ to $X \to Spec(\mathbb{Z})$, and so I thought it would be better to have the table exhibit this.

Maybe I should try to have a more pronounced horizontal rule in the middle where the table switches from the affine/projective line to its coverings? (Not sure how to do that though, but I might invent some hack.)

• CommentRowNumber8.
• CommentAuthorUrs
• CommentTimeAug 29th 2014

Earlier today I had given zeta function a considerably expanded Idea-section (effectively replacing the previous stubby Idea-section). It is explicitly anachronistic in that it presents the idea by going pretty much entirely opposite to the historical development of zeta functions. I believe this is the good way to actually explain the meaning of zeta functions.

• CommentRowNumber9.
• CommentAuthorTodd_Trimble
• CommentTimeAug 29th 2014

This is almost certainly going to be bewildering to people who do not speak the language of the Idea section.

The notion of zeta function is by now quite broad. But a good test case for the ideas of the Idea section might be the zeta function of a function field (with links to zeta functions for dynamical systems, etc.). I would prefer that the ideas of the Idea section be developed more at leisure, and somewhere further down the page, so that fewer people get lost immediately, and also to retain some sense of the history in the Idea section. This is just my personal opinion, of course.

• CommentRowNumber10.
• CommentAuthorUrs
• CommentTimeAug 29th 2014
• (edited Aug 29th 2014)

The zeta function of a function field, and of dynamical systems etc. are all mentioned further down. The idea was that it’s actually helpful not to start out with these arithmetic objects, because the actual meaning of zeta functions (beyond the collection of symbols that represents them on paper) is much clearer (or clear at all) for zeta functions of elliptic differential operators.

Of course if you say this doesn’t work for you, then something else must be done.

• CommentRowNumber11.
• CommentAuthorTodd_Trimble
• CommentTimeAug 29th 2014

This is all very interesting. I should probably consider more carefully how seriously my objection should be taken. But shooting from the hip: the exposition is highly condensed and concentrated (and in fact spread out over many nLab pages, which requires the ingenue to keep maybe twenty tabs open at once).

For what it’s worth: you’re right, it doesn’t quite work for me now. With some effort on my part that could change, but my current impression is that it would help immensely to develop this point of view in a more expository fashion, independently of the table. It’s obviously a great story to be told. Let me think further on how this criticism could be developed more constructively.

• CommentRowNumber12.
• CommentAuthorUrs
• CommentTimeSep 1st 2014
• (edited Sep 1st 2014)

I have put (in the Idea-section at zeta function) the mentioning of “Feynman propagators” in parenthesis, guessing that the appearance of this term is what caused part of the bewilderment. It’s however just a word for the analytic continuation of $Tr(H^{-s})$, which is what the entry really starts with.

I don’t think reading that Idea-section in its present form requires reading a bunch of other entries, certainly that is not the intention. All it really means to do is to instead of throwing some series expression at the reader – which is the way its usually done, but which doesn’t really give any “Idea”, I’d think – start by saying that we are looking at what in differential geometry are simply analytic continuations of $Tr(H^{-s})$ and that under the pertinent analogies the number-theoretic incarnations of zeta function may be thought of as being the arithmetic-geometry-incarnations of just this.

I don’t see any other way to really give an Idea (instead of just making the reader get used to) of what zeta-, theta-, eta-, and L-functions really are.