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Proportion of critical-line zeros of the Riemann zeta function

Let N(T)N(T) denote the number of nontrivial zeros ρ=β+iγ\rho=\beta+i\gamma of the Riemann zeta function ζ(s)\zeta(s) with

0<γ≤T,0<\gamma\le T,

counted with multiplicity. Let N0(T)N_0(T) count those zeros satisfying β=12\beta=\frac12, also with multiplicity.

Define

κ=lim inf⁡T→∞N0(T)N(T).\kappa=\liminf_{T\to\infty}\frac{N_0(T)}{N(T)}.

Problem. Determine κ\kappa. In particular, is

κ=1?\kappa=1?

The conjectured equality is equivalent to

N(T)−N0(T)=o(N(T)),N(T)-N_0(T)=o(N(T)),

meaning that asymptotically 100100% of the nontrivial zeros lie on the critical line.

This assertion is weaker than the Riemann hypothesis. As long as their proportion tends to zero, it permits finitely many, or even infinitely many, off-line zeros. The Riemann hypothesis permits no off-line nontrivial zeros.

Known progress. Selberg proved that κ>0\kappa>0. Conrey established κ>2/5\kappa>2/5, and Pratt, Robles, Zaharescu, and Zeindler subsequently proved κ>5/12\kappa>5/12.

The 2026 work of Alpöge and Furman, with a subsequent alternative proof by Lamzouri, establishes

κ>0.6725...\kappa>0.6725...

for zeros that are both simple and on the critical line.

References

References

J. B. Conrey, “More than two fifths of the zeros of the Riemann zeta function are on the critical line,” J. Reine Angew. Math. 399 (1989), 1–26.

K. Pratt, N. Robles, A. Zaharescu, and D. Zeindler, “More than five-twelfths of the zeros of ζ\zeta are on the critical line,” Research in the Mathematical Sciences 7 (2020), Article 2.

L. Alpöge and R. Furman, “More than two thirds of the zeta zeros are simple and on the critical line,” arXiv:2608.13637 (2026). https://arxiv.org/abs/2608.13637

Y. Lamzouri, “A new proof that more than 2/32/3 of the zeros of the Riemann zeta function are simple and on the critical line,” arXiv:2609.02882 (2026).

analytic-number-theoryasymptotic-densitycritical-line-zerosriemann-hypothesiszeta-functions
Posted by Austen F Posted · edited

Progress summary

Refreshed
Claimed progress

A 2018 paper claims that every nontrivial zero is asymptotically on the critical line, while newer work claims only a substantially stronger lower bound, so the problem remains unresolved.

The problem asks whether the proportion κ\kappa of nontrivial zeta zeros on the critical line equals 11. No claim here has received independent verification.

Known results

  • Selberg proved κ>0\kappa>0.
  • Conrey obtained κ>0.4088\kappa>0.4088.
  • Bui, Conrey, and Young reported κ≥0.4105\kappa\ge 0.4105 (2011).
  • Later work reported κ>0.410918\kappa>0.410918 (2014), while the stated subsequent record is κ>5/12\kappa>5/12.

2018 claim and 2026 claimed advance

In May 2018, arXiv:1805.07741 stated the theorem κ=1\kappa=1, but no verification or referee confirmation was found. A document found in September 2026 instead claims unconditionally that at least 0.67250…0.67250\ldots of zeros are simple and on the critical line, implying κ≥0.67250…\kappa\ge0.67250\ldots; it explicitly does not prove κ=1\kappa=1. Its publication date and authors are not shown.

Current status (as of September 2026): κ=1\kappa=1 is claimed but unverified; the strongest reported unconditional claim is κ≥0.67250…\kappa\ge0.67250\ldots, and the exact value remains open.

Sources

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