arXiv subjects

Daniel Alan Goldston

Publications and source records attributed to Daniel Alan Goldston.

4 recordsLinked to original sources

arXiv2026

Pair Correlation of Zeros of the Riemann Zeta Function I: Proportions of Simple Zeros and Critical Zeros

Assuming the Riemann Hypothesis (RH), Montgomery proved a theorem in 1973 concerning the pair correlation of zeros of the Riemann zeta-function and applied this to prove that at least 2/3 of the zeros are simple. In this paper, we investigate the versatility of the pair correlation method and show, for the first time, that it can be used to prove results on the horizontal distribution of zeros of the Riemann zeta-function. In earlier work we showed how to remove RH from Montgomery's theorem and, in turn, obtain results on simple zeros assuming conditions on the zeros that are weaker than RH. Here we assume a more general condition, namely that all the zeros $ρ= β+iγ$ with $T<γ\le 2T$ are in a narrow vertical box centered on the critical line with width $b/\log T$. We prove that under this assumption with $b=0.3185$ that at least $2/3$ of zeros are simple and on the critical line.

math.NT↗

arXiv2026

Pair Correlation Conjecture for the Zeros of the Riemann Zeta-function I: Simple and Critical Zeros

Montgomery in 1973 introduced the Pair Correlation Conjecture (PCC) for zeros of the Riemann zeta-function. He also conjectured that asymptotically 100% of the zeros are simple. His reasoning to support these two conjectures used the Riemann Hypothesis (RH). Building on Montgomery's approach, Gallagher and Mueller proved in 1978 that PCC under RH implies that 100% of the zeros are simple. Actually, the method of Gallagher and Mueller does not depend on RH, and thus Montgomery's second simplicity conjecture follows unconditionally from his PCC conjecture. We clarify this result by explicitly not assuming RH and considering PCC as a conjecture only concerning the vertical distribution of zeros. We then show that, for the first time, PCC can also be used to obtain information on the horizontal distribution of zeros. Using Gallagher and Mueller's method and a new idea concerning "horizontal multiplicity", we use PCC to prove that asymptotically 100% of the zeros are not only simple but also on the critical line.

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arXiv2025

The Alternative Hypothesis for Zeros of the Riemann Zeta-Function

In 2016, the first-named author introduced a formulation of the Alternative Hypothesis that assumes that consecutive zeros of the Riemann zeta-function are spaced at multiples of half of the average spacing, but does not assume that the zeros are simple. In this paper, we assume the Riemann Hypothesis and a similar formulation of the Alternative Hypothesis, and for each integer $k$ we obtain constraints on the density of pairs of zeros whose normalized differences are at $k/2$ times the average spacing. These constraints, in turn, restrict the density of (possible) multiple zeros. We also formulate a stronger version of the Alternative Hypothesis and show that it implies the Essential Simplicity Hypothesis.

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arXiv2023

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: Assuming all the zeros $ρ=β+iγ$ of the Riemann zeta-function such that $T^{3/8}<γ\le T$ satisfy $|β-1/2|<1/(2\log T)$, %lie in the thin box $\{s=σ+it: |σ-1/2|<1/(2\log T),\ T^{3/8}<t\le T\}$, 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 on or 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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