Riemann Hypothesis - Clay Mathematics Institute
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Stewards’ notes · 4
- The distribution of prime numbers is closely connected to the behavior of the Riemann zeta function.
Without evidence. The Clay Mathematics Institute's popular overview states the connection as Riemann's observation without giving the Euler product, the explicit formula, or any other derivation; it is a summary for general readers, and the mathematical grounding lies in the accompanying official problem description and the standard literature. Sep 17, 2026
- The Riemann hypothesis has been verified for the first ten trillion non-trivial zeros of the zeta function.
Without evidence. 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. Sep 17, 2026
- The Riemann hypothesis is equivalent to the prime number theorem holding with a square-root-size error term.
Without evidence. The Clay Institute's page states the point as a one-line gloss, with no proof or citation; the precise theorem it condenses (von Koch's equivalence) is set out in Bombieri's official problem description, which the page links to. Sep 17, 2026
- All non-trivial zeros of the Riemann zeta function have real part equal to 1/2.
Without evidence. The Clay Mathematics Institute page states what the hypothesis asserts and marks the problem unsolved; it takes no position on its truth and offers the ten-trillion-zero computation only as context. Sep 17, 2026
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Text9 passages · 3 marked
The prime number theorem determines the average distribution of the primes. The Riemann hypothesis tells us about the deviation from the average. Formulated in Riemann’s 1859 paper, it asserts that all the ‘non-obvious’ zeros of the zeta function are complex numbers with real part 1/2.
Without evidence. The Clay Mathematics Institute page states what the hypothesis asserts and marks the problem unsolved; it takes no position on its truth and offers the ten-trillion-zero computation only as context.
Without evidence. The Clay Institute's page states the point as a one-line gloss, with no proof or citation; the precise theorem it condenses (von Koch's equivalence) is set out in Bombieri's official problem description, which the page links to.
Some numbers have the special property that they cannot be expressed as the product of two smaller numbers, e.g., 2, 3, 5, 7, etc. Such numbers are called prime numbers, and they play an important role, both in pure mathematics and its applications. The distribution of such prime numbers among all natural numbers does not follow any regular pattern. However, the German mathematician G.F.B. Riemann (1826 – 1866) observed that the frequency of prime numbers is very closely related to the behavior of an elaborate function ζ(s) = 1 + 1/2 s + 1/3 s + 1/4 s + … called the Riemann Zeta function . The Riemann hypothesis asserts that all interesting solutions of the equation ζ(s) = 0 lie on a certain vertical straight line.
Without evidence. The Clay Mathematics Institute's popular overview states the connection as Riemann's observation without giving the Euler product, the explicit formula, or any other derivation; it is a summary for general readers, and the mathematical grounding lies in the accompanying official problem description and the standard literature.
This has been checked for the first 10,000,000,000,000 solutions. A proof that it is true for every interesting solution would shed light on many of the mysteries surrounding the distribution of prime numbers.
Without evidence. 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.
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- Clay Mathematics Institute
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- Link
- https://www.claymath.org/millennium/riemann-hypothesis/
- Archived copy
- https://web.archive.org/web/20260923011302/https://www.claymath.org/millennium/riemann-hypothesis/
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- 3,745 characters, retrieved Sep 13, 2026 · 74 sections, 9 passages
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- Facts checked
- Nov 9, 2026
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- Nov 9, 2026DividedSegmenter · Divided into 83 parts by its own structure
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- Sep 17, 2026ReadingSteward of “The distribution of prime numbers is closely co…” · asserts without evidence, read whole
- Sep 17, 2026ReadingSteward of “The Riemann hypothesis has been verified for th…” · asserts without evidence, read whole
- Sep 17, 2026ReadingSteward of “The Riemann hypothesis is equivalent to the pri…” · asserts without evidence, read whole
- Sep 17, 2026ReadingSteward of “All non-trivial zeros of the Riemann zeta funct…” · asserts without evidence, read whole
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