Montgomery's Pair Correlation Conjecture

Montgomery's Pair Correlation Conjecture

Definitions and mathematical statement

List the positive ordinates $\gamma$ of nontrivial zeta zeros with their multiplicities, and write $N(T)=\#\{0<\gamma\le T\}$. A standard fixed-separation form of the conjecture is, for every $0<a<b$, \[ \frac{1}{N(T)}\#\left\{0<\gamma,\gamma'\le T: a\le\frac{\log T}{2\pi}(\gamma-\gamma')\le b\right\} \longrightarrow\int_a^b\left(1-\left(\frac{\sin\pi u}{\pi u}\right)^2\right)du. \] Only positive separation is counted, so diagonal terms are absent. The scale $2\pi/\log T$ is the mean vertical spacing near height $T$. This is a pair-statistics law, not a claim that consecutive spacings have this density.

Formulation and concept references: [S1].

Short English statement

Do pairs of zeta zeros have the universal repulsion statistics predicted by the sine-kernel law?

Research attempt

Current frontier and quantifiers.

The source's Section 4 was inspected: the target concerns ordinates and does not assume RH. Goldston--Lee--Schettler--Suriajaya show that their PCC implies density-one simplicity and density-one location on the critical line, rather than RH for every zero [E1]. Restricted Fourier-support theorems are the foundational starting point, but the target tests every fixed positive interval. The attempt below isolates the analytic information lost by restricting Fourier support. It does not conflate pair statistics with consecutive gaps.

Attack route.

An explicit-formula route needs new control of off-diagonal prime sums. A positivity route hopes that the known low-frequency information fixes the whole pair law. A measure-convergence route asks for a uniform tail estimate that upgrades Fourier tests to interval counts. The second route is actively falsified below; the third is then stated as a precise sufficient package. No unproved prime correlation estimate is smuggled into that package as an established bound.

Attempt: the exact Fourier measure.

Let $c_T=\log T/(2\pi)$ and include all ordered pairs to form \[ \nu_T=\frac1{N(T)}\sum_{0<\gamma,\gamma'\leq T} \delta_{c_T(\gamma-\gamma')}. \] At zero this measure contains diagonal and repeated-ordinate contributions. For a Schwartz test function $h$, with $\widehat h(\xi)=\int h(u)e^{-2\pi i\xi u}\,du$, finite summation and Fourier inversion give \begin{equation}\label{wsref-0-65-71-37-34-3a-66-6f-75-72-69-65-72} \int h\,d\nu_T=\int_{{\mathbb{R}}}\widehat h(\xi)F_T(\xi)\,d\xi, \qquad F_T(\xi)=\frac1{N(T)} \left|\sum_{0<\gamma\leq T}e^{2\pi i\xi c_T\gamma}\right|^2. \end{equation} The sign of $\xi$ is immaterial because $F_T$ is even. In particular $F_T\geq0$, but its trivial pointwise bound $N(T)$ grows with $T$. The target itself asks only for convergence of $\nu_T([a,b])$ when $0<a<b$; it does not by definition specify the zero atom.

A sufficient, possibly stronger, full-measure candidate is \[ \nu_\infty=\delta_0+g(u)\,du, \qquad g(u)=1-\left(\frac{\sin\pi u}{\pi u}\right)^2. \] Its Fourier transform is the positive tempered measure $\delta_0+\min(|\xi|,1)\,d\xi$, since the Fourier transform of the squared sinc is $(1-|\xi|)_+$. Stating this sufficient candidate does not assume that the original interval formulation alone has already supplied all necessary tightness and zero-mass information.

Attempt: low-frequency information does not determine $g$.

Fix any finite frequency cutoff $A>0$. Choose a nonzero even real smooth function $b$ of compact support entirely in $|\xi|>A+1$. Let $\psi$ be its inverse Fourier transform and set $h(u)=u^2\psi(u)$. Then $h$ is real, even and Schwartz, and $\widehat h=-b''/(4\pi^2)$ also vanishes on $[-A,A]$. Near zero, \[ g(u)=\frac{\pi^2}{3}u^2+O(u^4),\qquad h(u)=O(u^2). \] Away from zero, $g$ is strictly positive; at infinity it tends to one while $h$ decays. Therefore $|h(u)|\leq K g(u)$ for some finite $K$. For $0<\varepsilon<1/K$, both \begin{equation}\label{wsref-0-65-71-37-34-3a-70-65-72-74-75-72-62} \delta_0+(g+\varepsilon h)\,du, \qquad \delta_0+(g-\varepsilon h)\,du \end{equation} are positive measures. They agree on every Schwartz test whose Fourier transform is supported in $[-A,A]$, by Fourier inversion and the support condition on $\widehat h$. They differ on some interval in $(0,\infty)$: $h$ is not identically zero there, so continuity gives an interval with nonzero integral.

Even positivity of the Fourier transform can be retained. On the support of $\widehat h$ the candidate Fourier density is one, and $\widehat h$ is bounded. Reducing $\varepsilon$ so that $\varepsilon\|\widehat h\|_\infty<1$ makes $\delta_0+[\min(|\xi|,1)\pm\varepsilon\widehat h(\xi)]d\xi$ positive as well. Thus positivity on both sides and any one fixed Fourier window cannot by themselves determine the sine-kernel pair measure. This is an analytic countertest to a uniqueness argument. The perturbed measures are not claimed to be pair measures of point processes, much less of zeta zeros.

Attempt: a sufficient tail-control upgrade.

Suppose the following two additional assertions held for the actual $F_T$. First, for every smooth compactly supported $\varphi$, \begin{equation}\label{wsref-0-65-71-37-34-3a-6c-6f-63-61-6c} \int\varphi(\xi)F_T(\xi)\,d\xi\longrightarrow \varphi(0)+\int\varphi(\xi)\min(|\xi|,1)\,d\xi. \end{equation} Second, for fixed constants $B,C$ independent of $T$ and every integer $j$, \begin{equation}\label{wsref-0-65-71-37-34-3a-74-61-69-6c} \int_j^{j+1}F_T(\xi)\,d\xi\leq C(1+|j|)^B. \end{equation} Then (eq74:local) extends from compactly supported functions to Schwartz functions. To prove this, multiply a Schwartz function by a smooth cutoff equal to one on $[-R,R]$. Its omitted integral is bounded, uniformly in $T$, by $C\sum_{|j|\geq R-1}(1+|j|)^B\sup_{[j,j+1]}|\varphi|$. Rapid decay makes this tail tend to zero. The limiting measure has the same property, so taking $T\to\infty$ first and then $R\to\infty$ is justified. Equation (eq74:fourier) now proves convergence on every smooth compactly supported $h$ in physical space.

For $0<a<b$, choose smooth minorants and majorants for $\mathbf1_{[a,b]}$ supported away from zero, whose differences have arbitrarily small integral against the continuous density $g$. Positivity of $\nu_T$ sandwiches the liminf and limsup. Sending the approximation error to zero gives the required interval limit. Thus the two displayed assertions suffice for the exact stated conclusion.

Stress test.

The missing part of (eq74:local) is precisely the unrestricted frequency range. A theorem for one fixed window is not the quantified assertion for every compact support. The perturbations (eq74:perturb) show why no positivity shortcut removes this gap. Likewise $F_T\leq N(T)$ does not supply (eq74:tail) with constants independent of height, so dominated convergence cannot be invoked with that bound. The sharp cutoff in $\nu_T$, any smoothed zero weights used in an explicit formula, and the absence of an RH assumption must all be matched before importing a Fourier theorem.

The full-measure sufficient package includes a unit zero atom, stronger information than merely declaring that diagonal pairs are excluded from $[a,b]$. It has not been extracted for free from the catalogue's statement. Nor does the density $g$ describe consecutive spacings: each zero is paired with every other zero. The argument neither computes zeros nor infers a limiting law from a finite plot.

Outcome.

Outcome: Exploratory approach with unresolved gap.

An explicit positive, positive-Fourier perturbation disproves the proposed uniqueness inference from low-frequency data. A proved measure-theoretic upgrade identifies sufficient full-frequency convergence and uniform tail control for the actual interval target. Those analytic inputs are not established. No pair-correlation theorem, zero-location theorem, or counterexample for zeta itself is obtained; the new reasoning is a methodological test without a priority claim.

Sources

• S1 Goldston, Lee, Schettler, and Suriajaya, Pair Correlation Conjecture for the zeros of the Riemann zeta-function I (version 4, 2026), Section 4, PCC. https://arxiv.org/html/2503.15449v4. Catalogue formulation source.
• S2 Variations of the Hardy Z-Function and the Montgomery Pair Correlation Conjecture. https://arxiv.org/abs/2511.18275. Catalogue research/status reference; not a proof endorsement.
• S3 Pair correlation of zeros of Dirichlet L-functions: a possible path towards the conjectures of Chowla, Elliott-Halberstam and Montgomery. https://link.springer.com/article/10.1007/s00208-026-03383-y. Catalogue research/status reference; not a proof endorsement.
• E1 Daniel A. Goldston, Junghun Lee, Jordan Schettler and Ade Irma Suriajaya, Pair Correlation Conjecture for the zeros of the Riemann zeta-function I: simple and critical zeros, arXiv:2503.15449v4, Section 4 and Theorem 1. https://arxiv.org/html/2503.15449v4. Inspected 27 September 2026. The source's PCC includes intervals approaching zero through its stated counting formulation; the present attempt keeps the supplied positive-interval target and uses a separately labelled sufficient full-measure package.