Math Conjectures

Montgomery’s Pair-Correlation Conjecture

OPENLandmarkConjectureProposed 1973 · Standard version

Canonical statement

Assume the Riemann hypothesis, write the nontrivial zeros of the Riemann zeta function as 12+iγ\tfrac12+i\gamma, counted with multiplicity, and put N(T)=#{γ:0<γ≤T}N(T)=\#\{\gamma:0<\gamma\le T\}. For every fixed 0<α<β0<\alpha<\beta,
1N(T)# ⁣{(γ,γ′):0<γ,γ′≤T,α≤(γ−γ′)log⁡T2π≤β}⟶∫αβ(1−(sin⁡πuπu)2)du \frac{1}{N(T)} \#\!\left\{(\gamma,\gamma'): 0<\gamma,\gamma'\le T, \alpha\le\frac{(\gamma-\gamma')\log T}{2\pi}\le\beta \right\} \longrightarrow \int_\alpha^\beta \left(1-\left(\frac{\sin \pi u}{\pi u}\right)^2\right)du
as T→∞T\to\infty.
View source LaTeX
Assume the Riemann hypothesis, write the nontrivial zeros of the Riemann zeta function as \(\tfrac12+i\gamma\), counted with multiplicity, and put \(N(T)=\#\{\gamma:0<\gamma\le T\}\). For every fixed \(0<\alpha<\beta\),
\[
 \frac{1}{N(T)}
 \#\!\left\{(\gamma,\gamma'):
   0<\gamma,\gamma'\le T,
   \alpha\le\frac{(\gamma-\gamma')\log T}{2\pi}\le\beta
 \right\}
 \longrightarrow
 \int_\alpha^\beta
   \left(1-\left(\frac{\sin \pi u}{\pi u}\right)^2\right)du
\]
as \(T\to\infty\).
NT-ZERO-034:[Montgomery1973PairCorrelation][GoldstonMontgomery1987PairCorrelation][Conrey2003RH]
Number theoryAnalysis and functional analysis

Notes

Montgomery proved a smoothed pair-correlation theorem when the Fourier transform of the test function has restricted support. Extending the result to unrestricted test functions, and hence to every fixed interval above, remains open; the conjecture is a finer zero-statistics assertion than RH itself.

References (3)

The boxed statement is the canonical open formulation — not a stronger variant or a related research program. The status reflects the catalog's last review; do your own literature search before investing serious effort.