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The Riemann hypothesis (open)

Bernhard Riemann

This page delivers no verdict; it presents claims and evidence with their sources. No theory in science is absolute. Method

For everyone

The most famous unproven conjecture, open since 1859, concerning the distribution of prime numbers. It has been checked by computer in trillions of cases with no counterexample, but in mathematics this does not count as a proof. It is one of the Clay Institute's Millennium Problems.

For specialists

The conjecture states that all non-trivial zeros of the Riemann zeta function lie on the critical line Re(s) = 1/2. This is equivalent to the error bound π(x) = Li(x) + O(√x log x) for the prime-counting function. Hardy (1914) proved that infinitely many zeros lie on the critical line. Platt and Trudgian verified the conjecture for all zeros with imaginary part up to 3·10¹². Littlewood's (1914) demonstration that π(x) − Li(x) changes sign is an example of how even extensive numerical evidence can mislead.

Claims

Evidence

Supporting

Computation / simulation

All zeros with imaginary part up to 3·10¹² lie on the critical line.[2]

Challenging

Methodological critique

Number theory has examples of patterns that seem to hold up to very large numbers and then fail at much larger values. Littlewood showed that π(x) − Li(x) changes sign infinitely often.[3]

Open questions

Connections

Parent

Mathematics

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Sources

  1. ArticleRiemann, B. (1859). Ueber die Anzahl der Primzahlen unter einer gegebenen Grösse. Monatsberichte der Königlich Preußischen Akademie der Wissenschaften zu Berlin, 671–680
  2. ArticlePlatt, D.; Trudgian, T. (2021). The Riemann hypothesis is true up to 3·10¹². Bulletin of the London Mathematical Society 53(3), 792–797DOI 10.1112/blms.12460archive
  3. ArticleLittlewood, J. E. (1914). Sur la distribution des nombres premiers. Comptes Rendus de l'Académie des Sciences 158, 1869–1872
  4. WebRiemann Hypothesis. Clay Mathematics Instituteclaymath.orgarchive

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Cite this page

APA

Athena contributors. (2026). The Riemann hypothesis (open). Athena: The tree of human knowledge. Retrieved November 1, 2026, from https://latifkedi.github.io/athena/en/d/riemann/

BibTeX
@misc{athena_riemann,
  title        = {The Riemann hypothesis (open)},
  author       = {{Athena contributors}},
  year         = {2026},
  howpublished = {\url{https://latifkedi.github.io/athena/en/d/riemann/}},
  note         = {Athena: The tree of human knowledge. Accessed: 2026-11-01}
}

Added: September 27, 2026 · Last changed: September 27, 2026 · Page history

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