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FrontierNumber theorysignificance 68

Proportion of zeta zeros on the critical line

The proportion κ\kappa of nontrivial zeros of ζ(s)\zeta(s) proved to lie on the line Re⁡ s=1/2\operatorname{Re}\, s = 1/2: N0(T)≥(κ−o(1)) N(T)N_0(T) \ge (\kappa - o(1))\, N(T).

Current best · higher is better
>67.25%> 67.25\%
Claude (unreleased research version) with Sumner, Alpöge, Furman and Easley, 10 Aug 2026 · entry
steps
6
by AI
1
since
1914
192019301940195019601970198019902000201020200%25%50%75%100%1914: infinitely many (Hardy)1942: a positive proportion (Selberg)1974: > 1/3 (Levinson)1989: > 2/5 (Conrey)2020: > 5/12 ≈ 41.7% (Pratt, Robles, Zaharescu and Zeindler)2026-08-10: > 67.25% (Claude (unreleased research version) with Sumner, Alpöge, Furman and Easley)

The line is the frontier over time. Filled dots are steps that moved it; muted dots are results that did not. Orange dots are catalog entries, results with AI in the loop. Hollow dots are candidates under review and never move the line. Grey dots along the bottom edge are results from before the quantity had a number, placed there because they have no value on this axis. Dots that would overlap are nudged sideways a few pixels. Hover a dot for its value and attribution.

About this frontier

The Riemann hypothesis says the proportion is 11. Hardy showed infinitely many zeros are on the line, Selberg a positive proportion, and from Levinson in 1974 the record climbed by mollifier refinements from a third to just over two fifths in half a century. The 2026 step is the first not to descend from Levinson's method and the first to pass one half.

Every step, newest first

DateValueWhoModelStatusSource
10 Aug 2026>67.25%> 67.25\%bestClaude (unreleased research version) with Sumner, Alpöge, Furman and EasleyClaude (unreleased research version)AI step
lean-checked
entry
2020>5/12≈41.7%> 5/12 \approx 41.7\%Pratt, Robles, Zaharescu and Zeindler–historicalsource ↗
1989>2/5> 2/5Conrey–historicalsource ↗
1974>1/3> 1/3
More than one-third of the zeros of Riemann's zeta function are on σ = 1/2.
Levinson–historicalsource ↗
1942a positive proportion
A small positive proportion, not made explicit.
Selberg–historicalsource ↗
1914infinitely many
Infinitely many zeros on the line; no proportion.
Hardy–historicalsource ↗

Historical rows follow the account in the Wikipedia article on the Riemann hypothesis (Hardy, Selberg, Levinson, Conrey, Pratt-Robles-Zaharescu-Zeindler) with Levinson's 1974 Advances in Mathematics title as the source of the one-third figure. Hardy's and Selberg's results have no proportion to plot and are recorded as ranks below Levinson.

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