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I've been reading the paper "LARGE SPACES BETWEEN THE ZEROS OF THE RIEMANN ZETA-FUNCTION" by S. H. Saker (arXiv-0906.5458v3 [math.NT]). http://arxiv.org/abs/0906.5458. The author employs conjectures on the moments of $\zeta(s)$ and $\zeta'(s)$ to derive bounds on the average spacing $\Lambda(k)$ between the zeros of $\zeta(s)$ on the critical line. Here $k$ refers to the order of the moment under consideration.

My question is this: what does it mean for the spacing to depend upon $k$? I'm sure I'm missing the point. I thought that there would be bounds that did not depend upon any parameter.

Thanks, Tom

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If you are really interested in this topic, I would suggest that you read R. R. Hall's papers on the subject. I have not looked at this paper, but ealier versions of Saker's papers on the arXiv contained many errors and were often based upon inequalities that were provably incorrect. Theorem 2.1 of this paper proves that $\Lambda>1.9902$ which is significantly worse that Hall's result that $\Lambda>2.63$. –  Micah Milinovich Sep 2 '10 at 19:43
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2 Answers 2

I believe $\Lambda(k)$ is the lower bound that arises from assuming the $k$th moment is predicted by random matrix theory (equation (1.13)) with the explicit constants $b(h,k)$ derived by Hughes. The method seems to require knowledge of the constants $b(k,k)$. Indeed at the top of page 5 he discusses the work of Hall in the case $h=3$, for which the value of $\Lambda$ derived (equation (1.17)) occurs in the table (1.18) as $\Lambda(3)$.

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I can't figure out what $\Lambda(k)$ is. The author defines $\Lambda$ by (1.4) on page 3, and it's an absolute constant. Then $\Lambda$ as a function shows up for the first time at (1.18) on page 6, with no explanation (that I can see) of what $\Lambda(3), \Lambda(4),\dots$ could mean.

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Thanks for the responses - QuantumBrian, I had gotten to the same understanding you mention in your response, but I still can't figure out what the meaning of the bound's dependence on $k$ is. It seems to me that a bound should be a bound! Maybe I need to ask Saker. John, I did have a look at one of Hall's papers, he set the stage for this work. Thanks again, Tom –  Tom Dickens Sep 11 '10 at 1:05
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