The Alternative Hypothesis for zeta-zero pairs: determinacy of the limiting densities p_{k/2} under AH-Pairs alone

The Alternative Hypothesis (AH) asserts, in the formulation studied by Goldston, Lee, Schettler, and Suriajaya (2025), that normalized differences between zeta zeros concentrate at half-integer multiples of the mean spacing, with densities $P_{k/2}(T)$ of pairs at $k/2$. That paper proved that, under AH-Pairs, the condition $p_0=1$ is equivalent to the Essential Simplicity Hypothesis (implying, without assuming the Riemann Hypothesis, that asymptotically 100% of zeta zeros are simple and on the critical line), but could establish $p_0=1$ only after adding averaged density hypotheses (AH-Weak Density). The open problem is the determinacy question this leaves open: do the limits $p_{k/2}=\lim_{T\to\infty}P_{k/2}(T)$ exist, and is $p_0=1$ forced, under AH-Pairs alone, without RH and without averaged-density hypotheses? The truth of AH itself — whether AH holds for the zeta zeros, given that Landau-Siegel zeros would force AH-type behavior while all currently provable band-limited correlation statistics are, by Lagarias and Rodgers, compatible with AH — is a related field-level question that the paper treats only as background; it is not part of the problem posed here. The determinacy question remains open.

Background

Let $\zeta(s)$ be the Riemann zeta-function. Its non-trivial zeros $\rho=\beta+i\gamma$ lie in the critical strip $0<\beta<1$, symmetrically with respect to the real axis and the critical line $\beta=1/2$; the Riemann Hypothesis (RH) asserts that all of them satisfy $\beta=1/2$. Writing $N(T)$ for the number of zeros with $0<\gamma\le T$, counted with multiplicity, the Riemann-von Mangoldt formula gives $N(T)\sim TL$ with $L=\frac{1}{2\pi}\log T$, so the mean vertical spacing between zeros at height $T$ is $1/L$. Montgomery's Pair Correlation Conjecture (PCC) predicts the two-point statistics of the normalized differences $(\gamma-\gamma')L$: for the pair-counting function $N(T,\lambda)$, which counts pairs of zeros with $0<(\gamma-\gamma')L\le\lambda$, PCC asserts $N(T,\lambda)=TL\int_0^\lambda\left(1-\left(\frac{\sin\pi\alpha}{\pi\alpha}\right)^2\right)d\alpha+o(TL)$. This sine-kernel form is characteristic of eigenvalues of random Hermitian matrices from the Gaussian Unitary Ensemble (GUE), the universality class that zeta zeros are widely believed to follow.

The Alternative Hypothesis (AH) is a rival picture of the same statistics: it says that normalized zero differences concentrate at half-integers rather than following a smooth density. It arose from arithmetic: a Landau-Siegel zero — an exceptionally close-to-$1$ real zero of a real Dirichlet $L$-function $L(s,\chi)$ (with $\chi$ a real Dirichlet character), whose possible existence is itself a famous open problem — would, as observed by Heath-Brown in a 1996 lecture, force $\zeta(s)$ to have infinitely many extremely long height ranges in which all zeros lie on the critical line, are simple, and are spaced at nearly exact integer multiples of half the average spacing. Abstracted from this arithmetic origin, that spacing property is the Alternative Hypothesis. It starkly contradicts PCC, yet it cannot currently be disproved: numerical computations of zeros (Odlyzko, 1987) strongly support PCC, but they cannot refute an asymptotic claim; Baluyot (2016) systematized the theory of AH, formulating it for consecutive zero spacings and deriving its consequences for pair correlation; and Lagarias and Rodgers (2020) proved that AH is compatible with every band-limited correlation statistic currently known for zeta zeros, by explicitly constructing a point process supported on the lattice $\frac{1}{2}\mathbb{Z}$ that satisfies AH and mimics all these statistics (they note that T. Tao independently obtained a similar construction).

The paper (Goldston, Lee, Schettler, Suriajaya, 2025) works with the pairs formulation AH-Pairs: for a fixed real number $M>0$ that may be taken arbitrarily large, every pair $(\gamma,\gamma')$ of zero ordinates in $P(T,M)=\{(\gamma,\gamma'):\; T/\log^2 T<\gamma,\gamma'\le T,\; |(\gamma-\gamma')L|\le M\}$ satisfies $(\gamma-\gamma')L=k/2+O((|k|+1)R(T))$ for some integer $k$, where $R(T)$ is a positive decreasing function tending to $0$. The density of pairs closest to $k/2$ is $P_{k/2}(T)=(TL)^{-1}\#\{(\gamma,\gamma')\in P(T,M):\; k/2-\tfrac14<(\gamma-\gamma')L\le k/2+\tfrac14\}$, and $p_{k/2}=\lim_{T\to\infty}P_{k/2}(T)$ when this limit exists. Two further hypotheses organize the subject. The Essential Simplicity Hypothesis (ESH, in the spirit of Mueller, 1983) asserts, roughly, that pairs of zeros whose ordinates $\gamma$ have equal height contribute density exactly one ($N^{\circledast}(T)=TL+o(TL)$, where $N^{\circledast}$ counts pairs with $\gamma=\gamma'$) together with short-range repulsion ($N(T,\lambda_0)=o(TL)$ as $\lambda_0\to0$); it is equivalent to the asymptotic essential simplicity of the zeros and implies that asymptotically 100% of zeros are simple and on the critical line. AH-Weak Density (conditions AH1 and AH2) postulates averaged values for the densities: $P_{j-1/2}(T)+P_j(T)=1-\frac{2}{\pi^2(2j-1)^2}+O(R_P(T))$ for each $j\ge1$, and $\sum_{j\le M}P_{j-1/2}(T)=\frac{M}{2}-\frac14+O(1/M)+O(MR_P(T))$, with $R_P(T)$ positive, decreasing, tending to $0$. The proof technique is the Gallagher-Mueller method, which compares the second moment $\int_0^T(N(t+\lambda/L)-N(t))^2dt$ of zero counts in short intervals with sums over zero pairs, using unconditional estimates on $S(t)=\frac{1}{\pi}\arg\zeta(\tfrac12+it)$, the fluctuating part of the zero-counting function; unlike Montgomery's original work this method does not require RH. Building on companion papers that proved an unconditional Montgomery theorem and showed that PCC implies ESH without RH (and hence that asymptotically 100% of zeros are simple and on the critical line), the paper proves: under AH-Pairs, $p_0=1$ is equivalent to ESH (Theorem 1); a long-averaged identity relating the densities $P_{k/2}(T)$ (Theorem 2); and, under AH-Pairs together with AH-Weak Density, $p_0=1$ and hence that asymptotically 100% of the zeros are simple and on the critical line, without RH (Theorem 4), while AH-Pairs with only condition AH1 yields $\limsup_{T\to\infty}P_0(T)\le 3/2$, at least 50% of zeros simple, and at least 50% of zeros on the critical line (Corollary 2).

Problem Statement

Work with the non-trivial zeros $\rho=\beta+i\gamma$ of the Riemann zeta-function, counted with multiplicity, where $N(T)\sim TL$, $L=\frac{1}{2\pi}\log T$, so the mean vertical spacing at height $T$ is $1/L$. Assume the Alternative Hypothesis for Pairs of Zeros (AH-Pairs): for each fixed real $M>0$ (which may be taken arbitrarily large) and every pair $(\gamma,\gamma')\in P(T,M)$, where

$P(T,M)=\{(\gamma,\gamma'):\; T/\log^2 T<\gamma,\gamma'\le T,\; |(\gamma-\gamma')L|\le M\},$

there is an integer $k$ with $(\gamma-\gamma')L=k/2+O((|k|+1)R(T))$ for some positive decreasing function $R(T)\to 0$; and let $P_{k/2}(T)=(TL)^{-1}\#\{(\gamma,\gamma')\in P(T,M):\; k/2-\tfrac14<(\gamma-\gamma')L\le k/2+\tfrac14\}$ for each integer $k$. Let the Essential Simplicity Hypothesis (ESH) be the conjunction of: (ES1) $N^{\circledast}(T)=TL+o(TL)$, where $N^{\circledast}(T)=\sum_{\rho,\rho':\,0<\gamma,\gamma'\le T,\;\gamma=\gamma'}1$ counts pairs of zeros with equal ordinates; and (ES2) $N(T,\lambda_0)=o(TL)$ whenever $\lambda_0\to 0$ as $T\to\infty$, where $N(T,\lambda_0)$ counts pairs of zeros with $0<\gamma,\gamma'\le T$ and $0<(\gamma-\gamma')L\le\lambda_0$. ESH is equivalent to $N^{\circledast}(T)+2N(T,\lambda_0)=TL+o(TL)$ and implies that asymptotically 100% of the zeros of $\zeta(s)$ are simple and on the critical line. The question below is to be answered without assuming the Riemann Hypothesis (RH) and without any additional averaged-density hypothesis — in particular without AH-Weak Density, i.e. without condition AH1 ($P_{j-1/2}(T)+P_j(T)=1-\frac{2}{\pi^2(2j-1)^2}+O(R_P(T))$ for each integer $j\ge1$, with $R_P$ positive, decreasing, tending to $0$) and without condition AH2 ($\sum_{j\le M}P_{j-1/2}(T)=\frac{M}{2}-\frac14+O(1/M)+O(MR_P(T))$ for large even $M$):

Determinacy under AH-Pairs alone: (i) does the limit $p_{k/2}=\lim_{T\to\infty}P_{k/2}(T)$ exist for each integer $k$; and (ii) is $p_0=1$ forced — equivalently, by Theorem 1 of the source paper, does AH-Pairs imply ESH, and hence that asymptotically 100% of the zeros of $\zeta(s)$ are simple and on the critical line? Or is AH-Pairs consistent with $p_0\ne1$?

References

1. Daniel A. Goldston, Junghun Lee, Jordan Schettler, Ade Irma Suriajaya, 'Pair Correlation Conjecture for the zeros of the Riemann zeta-function II: The Alternative Hypothesis', arXiv:2507.06823 (2025); published in Journal of Number Theory 291, 49-64, DOI 10.1016/j.jnt.2026.05.010, https://arxiv.org/abs/2507.06823
2. Siegfred Alan C. Baluyot, Daniel Alan Goldston, Ade Irma Suriajaya, Caroline L. Turnage-Butterbaugh, 'The Alternative Hypothesis for Zeros of the Riemann Zeta-Function', arXiv:2508.10857 (2025), https://arxiv.org/abs/2508.10857
3. Jeffrey C. Lagarias, Brad Rodgers, 'Higher Correlations and the Alternative Hypothesis', The Quarterly Journal of Mathematics 71 (2020), 257-280, DOI 10.1093/qmathj/haz043; arXiv:1905.12123, https://arxiv.org/abs/1905.12123
4. Siegfred Alan C. Baluyot, 'On the pair correlation conjecture and the alternative hypothesis', Journal of Number Theory 169 (2016), 183-226, DOI 10.1016/j.jnt.2016.05.007, https://doi.org/10.1016/j.jnt.2016.05.007
5. H. L. Montgomery, 'The pair correlation of zeros of the zeta function', Proceedings of Symposia in Pure Mathematics 24 (1973), 181-193, DOI 10.1090/pspum/024/9944, https://doi.org/10.1090/pspum/024/9944
6. P. X. Gallagher, Julia H. Mueller, 'Primes and zeros in short intervals', Journal für die reine und angewandte Mathematik 303/304 (1978), 205-220, DOI 10.1515/crll.1978.303-304.205, https://doi.org/10.1515/crll.1978.303-304.205
7. Julia Mueller, 'Arithmetic equivalent of essential simplicity of zeta zeros', Transactions of the American Mathematical Society 275 (1983), 175-183, DOI 10.1090/s0002-9947-1983-0678343-0, https://doi.org/10.1090/s0002-9947-1983-0678343-0
8. A. M. Odlyzko, 'On the distribution of spacings between zeros of the zeta function', Mathematics of Computation 48 (1987), 273-308, DOI 10.1090/S0025-5718-1987-0866115-0, https://doi.org/10.1090/S0025-5718-1987-0866115-0
9. D. R. Heath-Brown, 'Small class numbers and the pair correlation of zeros', talk at the AIM conference 'Riemann Hypothesis', Seattle, 1996, video archived at https://vimeo.com/showcase/4967015/video/287694279
10. Siegfred Alan C. Baluyot, Daniel Alan Goldston, Ade Irma Suriajaya, Caroline L. Turnage-Butterbaugh, 'An unconditional Montgomery theorem for pair correlation of zeros of the Riemann zeta-function', Acta Arithmetica 214 (2024), 357-376, DOI 10.4064/aa230612-20-3, https://doi.org/10.4064/aa230612-20-3
11. Siegfred Alan C. Baluyot, Daniel Alan Goldston, Ade Irma Suriajaya, Caroline L. Turnage-Butterbaugh, 'Pair Correlation of Zeros of the Riemann Zeta Function I: Proportions of Simple Zeros and Critical Zeros', arXiv:2501.14545 (2025), https://arxiv.org/abs/2501.14545
12. Daniel Alan Goldston, Junghun Lee, Jordan Schettler, Ade Irma Suriajaya, 'Pair Correlation Conjecture for the Zeros of the Riemann Zeta-function I: Simple and Critical Zeros', arXiv:2503.15449 (2025), https://arxiv.org/abs/2503.15449
13. Daniel A. Goldston, Ade Irma Suriajaya, 'Zeta Zeros on the Critical Line', arXiv:2511.20059 (2025), https://arxiv.org/abs/2511.20059