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ShowingImportancePrizesTopicRiemann hypothesis
The conjecture that all non-trivial zeros of the Riemann zeta function have real part equal to 1/2 (the critical line). Claims about its truth, status, proofs or attempted proofs, numerical verification, partial results, and consequences depend on it belong here; claims about the zeta function's other properties that do not concern the location of its zeros do not.
The distribution of prime numbers is closely connected to the behavior of the Riemann zeta function.
VerifiedThe claim traces to reliable primary sources through a clear chain of evidence.constitution →mathematicalA proposition of mathematics: true or false by proof rather than by observation. Settled by a proof others can check, and most firmly by one a machine has checked.constitution →Number theoryPrime number theoremimportance · settledImportance 0.15, from 0 to 1 · settled: uncontested, so low even when much depends on it. Higher-importance claims are worth more to assess, so funding reaches them sooner.constitution →
The Riemann hypothesis has been verified for the first ten trillion non-trivial zeros of the zeta function.
VerifiedThe claim traces to reliable primary sources through a clear chain of evidence.constitution →empirical · verifiableA factual claim that could be checked directly against observation or primary records.constitution →Number theoryimportance · settledImportance 0.18, from 0 to 1 · settled: uncontested, so low even when much depends on it. Higher-importance claims are worth more to assess, so funding reaches them sooner.constitution →
All non-trivial zeros of the Riemann zeta function have real part equal to 1/2.
SupportedEvidence favors the claim, but the chain is incomplete or the sources are secondary.constitution →mathematicalA proposition of mathematics: true or false by proof rather than by observation. Settled by a proof others can check, and most firmly by one a machine has checked.constitution →Number theoryimportance · majorImportance 0.80, from 0 to 1 · major: real consequence within a domain, actively argued. Higher-importance claims are worth more to assess, so funding reaches them sooner.constitution →
Gourdon's 2004 computation found all of the first 10^13 non-trivial zeros of the Riemann zeta function on the critical line.
UnassessedNo current assessment. Attention goes where its expected value is highest and someone funds it; nothing has funded an assessment of this claim yet, and anyone can.constitution →empirical · verifiableA factual claim that could be checked directly against observation or primary records.constitution →Number theoryimportance · settledImportance 0.15, from 0 to 1 · settled: uncontested, so low even when much depends on it. Higher-importance claims are worth more to assess, so funding reaches them sooner.constitution →
All non-trivial zeros of the Riemann zeta function up to height 3×10^12 lie on the critical line.
UnassessedNo current assessment. Attention goes where its expected value is highest and someone funds it; nothing has funded an assessment of this claim yet, and anyone can.constitution →empirical · verifiableA factual claim that could be checked directly against observation or primary records.constitution →Number theoryimportance · settledImportance 0.18, from 0 to 1 · settled: uncontested, so low even when much depends on it. Higher-importance claims are worth more to assess, so funding reaches them sooner.constitution →
The Riemann hypothesis is equivalent to the prime number theorem holding with a square-root-size error term.
VerifiedThe claim traces to reliable primary sources through a clear chain of evidence.constitution →mathematicalA proposition of mathematics: true or false by proof rather than by observation. Settled by a proof others can check, and most firmly by one a machine has checked.constitution →Number theoryPrime number theoremimportance · settledImportance 0.20, from 0 to 1 · settled: uncontested, so low even when much depends on it. Higher-importance claims are worth more to assess, so funding reaches them sooner.constitution →
Proving the Riemann hypothesis would substantially illuminate the distribution of the prime numbers.
UnassessedNo current assessment. Attention goes where its expected value is highest and someone funds it; nothing has funded an assessment of this claim yet, and anyone can.constitution →evaluativeA judgment of worth or quality against some standard: good, fair, effective.constitution →Number theoryimportance · minorImportance 0.30, from 0 to 1 · minor: narrow or largely settled, cheap to get right. Higher-importance claims are worth more to assess, so funding reaches them sooner.constitution →
The error term in the prime number theorem is O(x^θ log x) for every θ above the supremum of the real parts of the zeta zeros, and not O(x^θ) below it.
UnassessedNo current assessment. Attention goes where its expected value is highest and someone funds it; nothing has funded an assessment of this claim yet, and anyone can.constitution →mathematicalA proposition of mathematics: true or false by proof rather than by observation. Settled by a proof others can check, and most firmly by one a machine has checked.constitution →Prime number theoremNumber theoryimportance · settledImportance 0.15, from 0 to 1 · settled: uncontested, so low even when much depends on it. Higher-importance claims are worth more to assess, so funding reaches them sooner.constitution →
If the prime-counting function differs from the logarithmic integral by O(x^(1/2+ε)) for every ε > 0, then the Riemann hypothesis holds.
UnassessedNo current assessment. Attention goes where its expected value is highest and someone funds it; nothing has funded an assessment of this claim yet, and anyone can.constitution →mathematicalA proposition of mathematics: true or false by proof rather than by observation. Settled by a proof others can check, and most firmly by one a machine has checked.constitution →Number theoryPrime number theoremimportance · settledImportance 0.15, from 0 to 1 · settled: uncontested, so low even when much depends on it. Higher-importance claims are worth more to assess, so funding reaches them sooner.constitution →
If the Riemann hypothesis holds, the prime-counting function differs from the logarithmic integral by O(√x log x).
UnassessedNo current assessment. Attention goes where its expected value is highest and someone funds it; nothing has funded an assessment of this claim yet, and anyone can.constitution →mathematicalA proposition of mathematics: true or false by proof rather than by observation. Settled by a proof others can check, and most firmly by one a machine has checked.constitution →Prime number theoremNumber theoryimportance · settledImportance 0.15, from 0 to 1 · settled: uncontested, so low even when much depends on it. Higher-importance claims are worth more to assess, so funding reaches them sooner.constitution →
The Davenport-Heilbronn function satisfies a Riemann-type functional equation yet has zeros off the critical line.
UnassessedNo current assessment. Attention goes where its expected value is highest and someone funds it; nothing has funded an assessment of this claim yet, and anyone can.constitution →mathematicalA proposition of mathematics: true or false by proof rather than by observation. Settled by a proof others can check, and most firmly by one a machine has checked.constitution →Number theoryDavenport-Heilbronn functionimportance · settledImportance 0.15, from 0 to 1 · settled: uncontested, so low even when much depends on it. Higher-importance claims are worth more to assess, so funding reaches them sooner.constitution →
Michael Atiyah's 2018 claimed proof establishes the Riemann hypothesis.
UnassessedNo current assessment. Attention goes where its expected value is highest and someone funds it; nothing has funded an assessment of this claim yet, and anyone can.constitution →mathematicalA proposition of mathematics: true or false by proof rather than by observation. Settled by a proof others can check, and most firmly by one a machine has checked.constitution →Number theoryAtiyah's claimed Riemann hypothesis proofimportance · minorImportance 0.30, from 0 to 1 · minor: narrow or largely settled, cheap to get right. Higher-importance claims are worth more to assess, so funding reaches them sooner.constitution →
At least 40 percent of the non-trivial zeros of the Riemann zeta function lie on the critical line.
UnassessedNo current assessment. Attention goes where its expected value is highest and someone funds it; nothing has funded an assessment of this claim yet, and anyone can.constitution →mathematicalA proposition of mathematics: true or false by proof rather than by observation. Settled by a proof others can check, and most firmly by one a machine has checked.constitution →Number theoryimportance · settledImportance 0.15, from 0 to 1 · settled: uncontested, so low even when much depends on it. Higher-importance claims are worth more to assess, so funding reaches them sooner.constitution →
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