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I have noticed that there are quite a few publications, many of them recent, on trying to determine the supremum of the gaps (normalized) between zeros of $\zeta \left(\frac{1}{2} + i t \right)$. Several make use of Wirtinger's inequality. I've been studying some of these papers in an attempt to broaden my knowledge of the zeta function.

My question is this - why are people interested in determining information about the gaps? I assume that there must be some information encoded in the gap sizes about the distribution of prime numbers. Is there a publication or book in which I could read about the reasons for studying the gaps?

Thanks, Tom

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Hope this does not appear rude, it is not meant so: Don't the paper you read have some introduction? I am specifically asking as I just checked for some papers on this subject, and all of them contain some motivation (some more some less), basically along the lines of the first answer. –  quid Feb 19 '11 at 19:54
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No problem, not taken as rude. It turns out that the papers I've been reading (e. g. <arxiv.org/abs/0903.4007>;, <arxiv.org/abs/1001.0494>) don't address the motivation; I did some looking around and found a couple that do talk about the class number issues. Thanks for pointing this out. I'll need to widen my reading, and also learn about class numbers! Tom –  Tom Dickens Feb 20 '11 at 1:27

1 Answer 1

up vote 9 down vote accepted

Here are two papers that show a connection between the spacing of the zeros of the zeta function and the class number problem for imaginary quadratic fields:

Conrey, J. B., and H. Iwaniec, “Spacing of zeros of Hecke L-functions and the class number problem.”

Montgomery, H. L., and P. J. Weinberger, “Notes on small class numbers.”

And of course the study of the pair correlation of the ordinates of the zeros of the zeta-function inspired the connection between the zeta function and random matrix theory. I believe some results about the gaps are ways to check that the zeros are spaced according to predictions that come from random matrix theory.

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It is small gaps (not large) that are of interest in the connection to class number problem for imaginary quadratic fields. If it could be shown that a positive proportion of gaps have less than half the average spacing, then one could deduce an effective version of Siegel's bound for the class number. –  Micah Milinovich Feb 19 '11 at 5:15
    
Conrey was a student of Montgomery. –  Gerry Myerson Feb 19 '11 at 8:02

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