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The Alternative Hypothesis (AH) concerns a hypothetical and unlikely picture of how zeros of the Riemann zeta function are spaced, which one would like to rule out. In the Alternative Hypothesis, the renormalized distance between non-trivial zeros is supposed to always lie at a half integer. It is known that the Alternative Hypothesis is compatible with what is known about the pair correlation function of zeta zeros. We ask whether what is currently known about higher correlation functionsof the zeros is sufficient to rule out the Alternative Hypothesis and show by construction of an explicit counterexample point process that it is not. A similar result was recently independently obtained by Tao, using slightly different methods. We also apply the ergodic theorem to this point process to show there exists a deterministic collection of points lying in 1/2 Z, which satisfy the Alternative Hypothesis spacing, but mimic the local statistics that are currently known about zeros of the zeta function.
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An unconditional Montgomery theorem for pair correlation of zeros of the Riemann zeta-function
Assuming the Riemann Hypothesis (RH), Montgomery proved a theorem concerning pair correlation of zeros of the Riemann zeta-function. One consequence of this theorem is that, assuming RH, at least 67.9% of the nontrivial zeros are simple. Here we obtain an unconditional form of Montgomery’s theorem and show how to apply it to prove the following result on simple zeros: If all the zeros ρ=β+iγ of the Riemann zeta-function such that T3/8<γ≤T satisfy ∣∣β−1/2∣∣<1/(2logT), then, as T tends to infinity, at least 61.7% of these zeros are simple. The method of proof neither requires nor provides any information on whether any of these zeros are or are not on the critical line where β=1/2. We also obtain the same result under the weaker assumption of a strong zero-density hypothesis.
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- Award ID(s):
- 2239681
- PAR ID:
- 10529598
- Publisher / Repository:
- Institute of Mathematics Polish Academy of Sciences
- Date Published:
- Journal Name:
- Acta Arithmetica
- Volume:
- 214
- ISSN:
- 0065-1036
- Page Range / eLocation ID:
- 357 to 376
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.4064/aa230612-20-3
