---
title: Alternative Hypothesis (AH) in Zeta Zero Spacing
url: https://www.emergentmind.com/topics/alternative-hypothesis-ah
type: topic
---

# Alternative Hypothesis (AH) in Zeta Zero Spacing

In analytic number theory, the Alternative Hypothesis (AH) is a proposed spacing law for the nontrivial zeros of the Riemann zeta-function under which normalized zero-differences concentrate at half-integers rather than exhibiting the continuous spacing distribution predicted by the Gaussian Unitary Ensemble (GUE) or sine-kernel model. In its basic form, AH asserts that, after unfolding the ordinates so that the mean spacing is asymptotically \(1\), consecutive gaps are asymptotically \(1/2\)-integral; later work reformulates this as a statement about all zero-pairs, about limiting densities \(p_{k/2}\) at half-integer spacings, and about stronger variants with consequences for multiplicities and essential simplicity [1905.12123, 2508.10857].

## 1. Definition through normalization of zeta zeros

Assuming the Riemann Hypothesis (RH), write the nontrivial zeros as
\[
\rho_n=\tfrac12+i\gamma_n,\qquad 0<\gamma_1<\gamma_2<\cdots.
\]
The classical counting formula
\[
N(T)=\#\{0<\gamma_n\le T\} =\frac{T}{2\pi}\log\frac{T}{2\pi}-\frac{T}{2\pi}+O(\log T)
\]
implies that the average gap between consecutive ordinates near height \(T\) is
\[
\Delta(T)\sim\frac{2\pi}{\log T}.
\]
Accordingly, one introduces the normalized ordinates
\[
\widetilde\gamma_n=\frac{\gamma_n}{2\pi}\,\log\gamma_n,
\]
so that the average spacing between successive \(\widetilde\gamma_n\) is asymptotically \(1\) [2508.10857].

In the simpler spacing formulation, AH states that for all large \(j\),
\[
\widetilde\gamma_{j+1}-\widetilde\gamma_j=h_j+o(1),\qquad h_j\in\{\tfrac12,1,\tfrac32,2,\dots\}.
\]
Equivalently, apart from a vanishing error, every renormalized gap lies on a half-integer lattice. A more precise version, introduced in 2016, postulates a function \(\psi(\gamma)\to\infty\) with \(\psi(\gamma)=o(\log\gamma)\) such that for each \(n\ge1\) one can choose an integer \(k_n\ge0\) satisfying
\[
\widetilde\gamma_{n+1}-\widetilde\gamma_n
=\tfrac12\,k_n+O\Bigl((\gamma_{n+1}-\gamma_n)\,\psi(\gamma_n)\Bigr).
\]

The contrast with the standard GUE heuristic is sharp. Under the usual GUE conjecture, the unfolded zeros converge to a translation-invariant determinantal process with sine kernel
\[
S(x)=\frac{\sin(\pi x)}{\pi x},
\]
and normalized spacings occur on a continuum of scales. Under AH, by contrast, the asymptotic spacing support is discrete.

## 2. Equivalent pairwise and density formulations

Later work develops AH into a hierarchy of formulations centered on all zero-pairs rather than only consecutive ones. These formulations are not merely cosmetic: they are the forms used in pair-correlation arguments, density identities, and simplicity results [2508.10857, 2507.06823].

| Formulation | Statement | Role |
|---|---|---|
| **AH (consecutive gaps)** | \(\widetilde\gamma_{n+1}-\widetilde\gamma_n=\tfrac12 k_n+O((\gamma_{n+1}-\gamma_n)\psi(\gamma_n))\) | Original half-integer gap formulation |
| **AH2 / AH-Pairs** | For \((\gamma,\gamma')\in\mathcal P(T,M)\), \((\gamma-\gamma')\frac{\log T}{2\pi}=\frac{k}{2}+O((|k|+1)R(T))\) | Extends half-integer localization to all nearby pairs |
| **AH-Density** | \(p_{k/2}=\lim_{T\to\infty}P_{k/2}(T)\) exists and encodes the limiting mass at \(k/2\) | Produces a discrete pair-correlation law |
| **AH2 Strong** | Same pair localization on \(\mathcal Q(T,\mathcal M)=\{|\gamma-\gamma'|\le\mathcal M\}\) with \(R(T)\log T\to0\) | Strong enough to force \(p_0=1\) |
| **AH-Weak Density** | \(P_{j-1/2}(T)+P_j(T)=1-\frac{2}{\pi^2(2j-1)^2}+o(1)\) and \(\sum_{j=1}^M P_{j-1/2}(T)=\frac{M}{2}-\frac14+o(1)\) | Used in the unconditional Gallagher–Mueller argument |

For AH2, one introduces
\[
\mathcal P(T,M)=\bigl\{(\gamma,\gamma'):\tfrac{T}{\log^2T}<\gamma,\gamma'\le T,\ \bigl|\tfrac{\gamma-\gamma'}{2\pi}\log T\bigr|\le M\bigr\},
\]
partitions it into blocks
\[
B_{k/2}(T,M,\delta)
=\Bigl\{(\gamma,\gamma')\in\mathcal P(T,M):
\bigl|\tfrac{\gamma-\gamma'}{2\pi}\log T-\tfrac{k}{2}\bigr|\le\delta/2\Bigr\},
\]
and defines
\[
P_{k/2}(T)=\frac{1}{(T/2\pi)\log T}\,\bigl|B_{k/2}(T,M,\delta)\bigr|.
\]
AH-Density is the assertion that the limiting densities \(p_{k/2}\) exist. In the pair-correlation language of Goldston–Lee–Schettler–Suriajaya, the corresponding limiting distribution is the atomic measure
\[
F_{AH}(\alpha)=\sum_{k\in\mathbb Z}p_{k/2}\,\delta\!\bigl(\alpha-\tfrac{k}{2}\bigr),
\]
with
\[
\sum_{k\in\mathbb Z}p_{k/2}=1,\qquad p_{k/2}\ge0.
\]

## 3. Relation to GUE and to the known correlation theorems

The full GUE hypothesis predicts that for each \(n\ge1\) and Schwartz \(\phi\in S(\mathbb R^n)\),
\[
\lim_{T\to\infty}\frac1T\int_T^{2T}\sum_{j_1,\dots,j_n}
\phi(\widetilde\gamma_{j_1}-t,\dots,\widetilde\gamma_{j_n}-t)\,dt
=
\int_{\mathbb R^n}\phi(x)\det_{1\le i,j\le n}[S(x_i-x_j)]\,d^n x.
\]
Rigorously, however, this is known only for band-limited \(\phi\) whose Fourier transform is supported in
\[
\{\xi\in\mathbb R^n:|\xi_1|+\cdots+|\xi_n|<2\},
\]
and also away from the hyperplane \(\xi_1+\cdots+\xi_n=0\). Montgomery proved the \(n=2\) case in this band, Hejhal proved \(n=3\), and Rudnick–Sarnak treated all higher \(n\) in the same range [1905.12123].

This restriction is the central reason AH remains mathematically viable. It is known that AH is compatible with what is known about the pair-correlation function of zeta zeros, and Lagarias–Rodgers showed that currently known higher-correlation information is also insufficient to rule it out. Since every rigorously proved correlation estimate for the true zeta zeros fits into this band-limited category, any model reproducing those restricted statistics will pass all currently proved pair- and higher-correlation tests exactly.

A common misunderstanding is that agreement with the known sine-kernel correlation formulas already excludes half-integer spacing. The existing theorems do not justify that conclusion, because their Fourier-support range is too narrow. A plausible implication is that excluding AH requires either correlation estimates beyond these Fourier-support barriers or different statistics, such as information involving zeros of \(\zeta'\) or moment bounds.

## 4. Lagarias–Rodgers point process and the deterministic AH model

Lagarias and Rodgers constructed an explicit determinantal point process on the half-integer lattice \(\tfrac12\mathbb Z\) whose kernel is
\[
K(j,k)=S(j-k),\qquad j,k\in\tfrac12\mathbb Z,
\]
with \(S(x)=\sin(\pi x)/(\pi x)\). They then randomly translate this process by an independent \(\omega\in[0,\tfrac12)\), uniformly distributed, to produce an \(\mathbb R\)-stationary process \(\tilde w_j=w_j-\omega\) [1905.12123].

The construction is exact rather than heuristic. The underlying \( \tfrac12 \)-discrete sine process exists because its integral operator is the orthogonal projector onto the band \([-\tfrac12,\tfrac12]\), hence satisfies \(0\le K\le I\) and is trace-class on compacts. By Poisson summation, for any band-limited test function \(\phi\in S(\mathbb R^n)\) with \(\operatorname{supp}\hat\phi\subset[-1,1]^n\), the \(n\)-point averages of this translated lattice process coincide exactly with those of the continuum sine process:
\[
E\,\phi(\tilde w_{j_1},\dots,\tilde w_{j_n})
=
\int_{\mathbb R^n}\phi(x)\det[S(x_i-x_j)]\,d^n x.
\]

At the same time, every spacing in every realization belongs to
\[
\{\tfrac12,1,\tfrac32,\dots\}.
\]
Thus the process simultaneously satisfies AH spacing and reproduces all rigorously known correlation statistics.

They further apply the ergodic theorem to obtain a single deterministic sequence \(c_j\in\tfrac12\mathbb Z\) with the same property. The half-step shift action is ergodic, and Birkhoff’s theorem converts orbit averages into the required continuum shift averages. The resulting deterministic configuration satisfies
\[
c_{j+1}-c_j\in\{\tfrac12,1,\tfrac32,\dots\}
\]
for all \(j\), yet for all band-limited \(\phi\) its \(n\)-point correlation statistics agree with the sine-kernel predictions. This is Theorem 1.5 of the paper. A similar result was independently obtained by T. Tao using slightly different methods.

## 5. Density identities, multiple zeros, and essential simplicity

Assuming RH together with AH2, Baluyot–Goldston–Suriajaya–Turnage-Butterbaugh derive explicit constraints on the half-integer densities \(P_{k/2}(T)\) and their limits. Their main theorem gives
\[
1+o(1)\le P_0(T)\le \frac32-\frac{2}{\pi^2}+o(1),
\]
and for each nonzero \(k\in\mathbb Z\),
\[
P_{k/2}(T)\sim
\begin{cases}
P_0(T)-\tfrac12,& k\text{ even},\ k\ne0,\\[4pt]
\tfrac32-\tfrac{2}{\pi^2k^2}-P_0(T),& k\text{ odd}.
\end{cases}
\]
As a corollary, if one limit \(p_{k_0/2}\) exists, then all limits \(p_{k/2}\) exist and satisfy the same relations [2508.10857].

The quantity \(P_0\) has a direct multiplicity interpretation. The block \(B_0\) counts coinciding zeros: a simple zero contributes once, while a zero of multiplicity \(m\) contributes \(m^2\) times to the double sum. Hence \(P_0\) controls the mean square multiplicity. The bound
\[
P_0\le \tfrac32-\tfrac{2}{\pi^2}\approx1.29735
\]
shows that under AH, any limiting mean square multiplicity cannot exceed approximately \(1.297\); in particular, AH excludes overly frequent multiple zeros.

The same paper isolates a stronger formulation, AH2 Strong, in which pair localization holds uniformly on windows \(\mathcal Q(T,\mathcal M)=\{|\gamma-\gamma'|\le\mathcal M\}\) with \(R(T)\log T\to0\). Under RH plus AH2 Strong, one obtains
\[
p_0=\lim_{T\to\infty}P_0(T)=1,
\]
and therefore the Essential Simplicity Hypothesis (ESH) holds. In this formulation, ESH means that almost all zeros are simple and almost no two distinct zeros lie arbitrarily close; under RH it is equivalent to \(p_0=1\).

The same framework also reconstructs the pair-correlation function beyond the classical range. Under RH plus AH-Density, one gets an exact periodic “sawtooth + \(\delta\)-spikes at half-integers” description of \(F(\alpha)\), with
\[
\mathcal F(\alpha)=|\alpha|\quad(|\alpha|\le1),\qquad \mathcal F(\alpha+2)=\mathcal F(\alpha),
\]
supplemented by delta-masses of weight \(2(p_0-1)\) at odd integers and half-weights at \(0,\pm1\). This formalizes the sense in which AH provides a fully discrete alternative to the continuous GUE pair-correlation law.

## 6. Unconditional pair-correlation consequences and mathematical status

Goldston–Lee–Schettler–Suriajaya formulate AH directly at the level of pair correlation without assuming RH. With
\[
L=\frac{1}{2\pi}\log T
\]
and
\[
R_T(w)=\frac1{N(T)}\sum_{0<\gamma,\gamma'\le T}w\!\bigl((\gamma-\gamma')L\bigr),
\]
Montgomery’s Pair Correlation Conjecture predicts the continuous limit
\[
\int_{-\infty}^{\infty}\Bigl(1-\bigl(\tfrac{\sin\pi\alpha}{\pi\alpha}\bigr)^2\Bigr)w(\alpha)\,d\alpha.
\]
Under AH-Density, by contrast,
\[
R_T(w)\to \sum_{k\in\mathbb Z}p_{k/2}\,w(k/2),
\]
equivalently against the atomic kernel \(F_{AH}(\alpha)=\sum_{k\in\mathbb Z}p_{k/2}\delta(\alpha-k/2)\) [2507.06823].

Their argument imports the Gallagher–Mueller method through the triangularly weighted sum
\[
D(T,\lambda)=\sum_{0<\gamma,\gamma'\le T}\bigl(\Delta-|\gamma-\gamma'|\bigr)_+,\qquad \Delta=\frac{\lambda}{L},
\]
and combines an unconditional identity for
\[
\int_0^T (N(t+\Delta)-N(t))^2\,dt
\]
with the AH decomposition into half-integer blocks. The resulting combinatorial identity, together with AH-Weak Density, implies
\[
P_0(T)=1+o(1).
\]
From this they deduce the Essential Simplicity Hypothesis and then the asymptotic conclusion
\[
\lim_{T\to\infty}\frac1{N(T)}
\#\{\gamma\le T:\rho=\beta+i\gamma\text{ is simple and }\beta=\tfrac12\}=1.
\]
Thus AH-Pairs plus AH-Weak Density imply that asymptotically \(100\%\) of the nontrivial zeros are simple and lie on the critical line, without any assumption of RH.

This places AH in a distinctive position. It is only a conjecture on pair correlations and does not by itself assert that every zero lies on \(\Re s=\tfrac12\). Nevertheless, in the Gallagher–Mueller framework it is as effective as Montgomery’s classical Pair Correlation Conjecture for proving that almost all zeros are simple and critical. At the same time, the broader evidentiary picture is unfavorable to AH: it can arise, for example, from postulating extremely small Landau–Siegel zeros of certain real Dirichlet \(L\)-functions, but the classical pair-correlation law predicts a continuous repulsion profile, and extensive numerical data support that continuous law and contradict AH [2507.06823]. The current state of the subject is therefore a precise tension: AH is mathematically compatible with all rigorously proved correlation data now available, yet it stands in conflict with the prevailing numerical and heuristic picture.

Source: https://www.emergentmind.com/topics/alternative-hypothesis-ah