The Riemann hypothesis has been verified for the first ten trillion non-trivial zeros of the zeta function.
Assessment
The claim traces to reliable primary sources through a clear chain of evidence.
The first ten trillion non-trivial zeros of the Riemann zeta function have been checked and all lie on the critical line. The figure originates in Xavier Gourdon's 2004 computation of the first 10^13 zeros, carried out with Patrick Demichel using the Odlyzko–Schönhage algorithm. That computation was never published in a refereed journal, and its handling of rounding and truncation error was questioned by later authors, so for some years the ten-trillion figure rested on an unrefereed report.
The doubt has since been removed. In 2021 Dave Platt and Tim Trudgian published, in the Bulletin of the London Mathematical Society, a rigorous verification using interval arithmetic and Turing's method that all zeros up to height 3×10^12 lie on the critical line; this covers the lowest 12,363,153,437,138 zeros, about 24 percent more than ten trillion, and independently confirms Gourdon's and Wedeniwski's earlier results. The claim as stated is therefore established by a refereed computation. It says nothing about the Riemann hypothesis itself, which remains open: finite verification to any height is inductive evidence only.
Full reasoning: the evidence and decisions behind this verdict
Two sources were read whole. The Clay Mathematics Institute problem page (www.claymath.org/millennium/riemann-hypothesis/) states that the hypothesis "has been checked for the first 10,000,000,000,000 solutions" without citing a computation; the figure matches Gourdon's 2004 count and is a restatement rather than independent evidence. The primary source is Platt and Trudgian, "The Riemann hypothesis is true up to 3·10^12" (arxiv.org/pdf/2004.09765; Bull. LMS 2021, doi:10.1112/blms.12460), whose Theorem 1 states that the hypothesis holds up to height 3,000,175,332,800, i.e. that the lowest 12,363,153,437,138 non-trivial zeros have real part 1/2. Their introduction reviews the prior record: Wedeniwski's ZetaGrid (2004, height about 2.41×10^11, with inconsistent counts across its own reports), Gourdon (2004, height about 2.44×10^12, the 10^13-zero computation), and Platt's own 2017 rigorous run to about 3.06×10^10. They note that neither Wedeniwski's nor Gourdon's result was refereed and that error accumulation in them is unclear, concerns also raised by Tao and Helfgott, and they state that their result independently verifies those computations and exceeds the largest by 22 percent.
Weighing: the subclaim that all zeros up to height 3×10^12 lie on the critical line entails the claim by itself, since 12.36 trillion exceeds ten trillion; it is a refereed result produced with rigorous ball arithmetic (Arb) and a Turing-method zero count, and no published objection to it is known. The subclaim that Gourdon's 2004 computation found all of the first 10^13 zeros on the critical line is the historical origin of the figure and is now confirmed by the independent computation, so its lack of peer review no longer weakens the claim. Both recorded instances affirm the claim; no source denies it. The residual uncertainty is the ordinary possibility of an undetected error in a large computation, small because two independent implementations agree over the whole range.
What would change the verdict: a published error in the Platt–Trudgian computation together with a failure of Gourdon's, or a zero found off the line below height 2.44×10^12, none of which has been reported.
Decomposition
The claims this one rests on directly. ↗︎ opens a subclaim; the map shows how they fit together.
The claims this one rests on directly, not gathered into a named line of reasoning.
- supportsthis provides evidence for the parentsteward instructions →All non-trivial zeros of the Riemann zeta function up to height 3×10^12 lie on the critical line. ↗︎
- supportsthis provides evidence for the parentsteward instructions →Gourdon's 2004 computation found all of the first 10^13 non-trivial zeros of the Riemann zeta function on the critical line. ↗︎
Provenance
Where this claim has been said, linked to its canonical form.
This has been checked for the first 10,000,000,000,000 solutions.
Referring to the interesting solutions of ζ(s) = 0 lying on the critical line.
Asserted without evidence of the source's own. The Clay Mathematics Institute page states the ten-trillion figure without citing any computation. The figure matches Gourdon's 2004 count of 10^13 zeros, and it has since been independently confirmed by the refereed computation of Platt and Trudgian, which reaches about 12.4 trillion zeros.
Theorem 1. The Riemann hypothesis is true up to height 3 000 175 332 800. That is, the lowest 12 363 153 437 138 non-trivial zeroes ρ have ℜρ = 1/2.
The paper's main theorem, established by a rigorous interval-arithmetic computation with Turing's method; it states a result that covers more than twelve trillion zeros and so entails the ten-trillion figure, and the authors note it independently verifies Gourdon's earlier 10^13-zero computation.
The source's own evidence bears what it asserts. Platt and Trudgian describe their method (Arb ball arithmetic, sign-change counting on the half line, a variant of Turing's method to certify that no zeros are missed) and the computational resources used, and they state the exact height and zero count. The paper appeared in the Bulletin of the London Mathematical Society in 2021. It is the primary source a reader should open for the current rigorous verification record. Worth reading closely: It is the refereed primary source establishing the verification for more than ten trillion zeros, and it explains why the earlier Gourdon and Wedeniwski computations were considered less than fully rigorous.
How these sources relate
- https://www.claymath.org/millennium/riemann-hypothesis/ restates Computation of zeros of the Zeta function (Gourdon and Demichel, 2004), faithfully. The Clay page restates the ten-trillion figure with no evidence of its own. The number coincides with Gourdon's 2004 count of 10^13 zeros, which was the record at the time the page text was written and is the figure the popular discourse repeats. The page names no source, so the dependency is inferred from the figure rather than from an explicit citation; the confidence reflects that. The restatement drops no qualification the upstream report carries, though the report itself was never refereed. Judged from the citing document alone.
Cite this claim: a formal citation with its evidence attached
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Created by extractor · Sep 13, 2026. Every judgment on this page is accompanied by a reasoning trace.