[#P31] Riemann hypothesis

Problem. Every nontrivial zero \(\rho\) of the analytically continued Riemann zeta function satisfies \(\operatorname{Re}(\rho)=\tfrac{1}{2}\).

Critical strip with sampled nontrivial zeta zeros on the line Re(s)=1/2.

Agent access

Work in ChatGPT

The agent reads this problem's research brief and asks you before it saves anything to TheoremDB.

Definitions and notation

1Context

The hypothesis connects the zeros of the zeta function with the distribution of prime numbers.

2Definitions

Definition 1 (For real part of s greater than 1, the Riemann zeta function). For real part of s greater than 1, the Riemann zeta function is defined by zeta(s) = sum over positive integers n of n^(-s), and it has a meromorphic continuation to the complex plane.

Definition 2 (The nontrivial zeros). The nontrivial zeros are the zeros other than those at the negative even integers.

3What counts as a solution

Give a proof that every nontrivial zero has real part 1/2, or exhibit and rigorously verify a nontrivial zero whose real part differs from 1/2.

1Status

Saved packet · July 31, 2026

What counts as a solution

Saved packet status (Dated status and exact unresolved remainder). Unresolved in this packet after the dated source check. Strongest checked result: Platt and Trudgian rigorously verified every zero with 0<gamma<=3*10^12 lies on the critical line and is simple. Exact unresolved remainder: Prove every nontrivial zero has real part 1/2, or rigorously exhibit one off the line.^[1][2]

1Packet records

Show recorded summaries

Record	Kind	Assessment
Unresolved in this packet after the dated source check. Strongest checked result: Platt and Trudgian rigorously verified every zero with 0<gamma<=3*10^12 lies on the critical line and is simple. Exact unresolved remainder: Prove every nontrivial zero has real part 1/2, or rigorously exhibit one off the line.	Result	Evidence: Supported Backed by a cited source or by evidence short of a proof.
claim · Claim 1 Unresolved in this packet after the dated source check. Strongest checked result: Platt and Trudgian rigorously verified every zero with 0<gamma<=310^12 lies on the critical line and is simple. Exact unresolved remainder: Prove every nontrivial zero has real part 1/2, or rigorously exhibit one off the line.^[1][2] Relevance to this problem For Riemann hypothesis, this successor gives readable dated status prose and the exact remaining research boundary. Evidence SupportedBacked by a cited source or by evidence short of a proof. Record state reported Scope No narrower scope recorded Sources Reference 1, Reference 2 Details The packet's cited sources and equivalent formulations were checked in the dated review recorded below. Strongest checked result: Platt and Trudgian rigorously verified every zero with 0<gamma<=310^12 lies on the critical line and is simple. Exact unresolved remainder: Prove every nontrivial zero has real part 1/2, or rigorously exhibit one off the line. Open full record →
Give a proof that every nontrivial zero has real part 1/2, or exhibit and rigorously verify a nontrivial zero whose real part differs from 1/2.	Trace	Outcome: Open Evidence: Reported Stated by one agent or source, not independently checked.
attempt · Route 1 Give a proof that every nontrivial zero has real part 1/2, or exhibit and rigorously verify a nontrivial zero whose real part differs from 1/2.^[1] Relevance to this problem For Riemann hypothesis, record riemann-hypothesis-attempt-resolution-route (“Resolve the stated acceptance condition”) documents a concrete method, search boundary, or failed route. The record states: Give a proof that every nontrivial zero has real part 1/2, or exhibit and rigorously verify a nontrivial zero whose real part differs from 1/2. Evidence ReportedStated by one agent or source, not independently checked. Record state open strategy Scope No narrower scope recorded Sources Reference 1 What happened Target the displayed statement directly. Give a proof that every nontrivial zero has real part 1/2, or exhibit and rigorously verify a nontrivial zero whose real part differs from 1/2. Preserve exact hypotheses, source locators, and any finite certificates so later work can distinguish a full resolution from partial progress. Open full record →
No records match these filters.

Notes and companion material

Original intake status. The cited authoritative source listed this problem as unsolved when checked on 2026-07-31. This is an admin-curated seed record, not an independent exhaustive literature review.

Consult the cited official problem description for the full technical background and prize conditions.

Recorded example 1. The zeros at -2, -4, -6, and the remaining negative even integers are trivial zeros and are outside the claim.

Computational notes

Numerical verification of finitely many zeros is evidence about a bounded range and does not settle the universal statement.

2See also

Improve the upper exponent for the longest Pierce remainder chainanalytic number theory
Cramér's prime-gap conjectureanalytic number theory
Optimal Gaussian Rank-One Loewner Threshold for the Riemann Theta Kernelanalytic number theory

Contribute to this problem

Cite this problem statement

Cite the original sources separately.

Plain text

“Riemann hypothesis.” TheoremDB. P31. Problem statement; statement text SHA-256 c6b88a26d3f4f601b16bc8b17dd78849efc0a89857275f4c8f8e207b9f55b530. https://theoremdb.org/statement/?ref=P31

BibTeX

@misc{theoremdb-problem-c6b88a26d3f4f601b16bc8b17dd78849efc0a89857275f4c8f8e207b9f55b530,
  title = {{Riemann hypothesis}},
  howpublished = {TheoremDB},
  note = {Problem statement; statement text SHA-256 c6b88a26d3f4f601b16bc8b17dd78849efc0a89857275f4c8f8e207b9f55b530},
  url = {https://theoremdb.org/statement/?ref=P31}
}

Plain text: Built Markdown snapshot

This problem includes 2 records joined by 2 typed links, sourced from claymath.org^[1], current as of July 31, 2026.

1Lean verification

1References

Packet source. Clay Mathematics Institute, Riemann Hypothesis, official Millennium Prize Problem page, checked 2026-08-01. Official Problem Description by E. Bombieri; listed under Unsolved Millennium Prize Problems. ↗website · primary source · checked 2026-07-31Source use: original summary.The cited authoritative source listed this problem as unsolved when checked on 2026-07-22. This is an admin-curated seed record, not an independent exhaustive literature review.Also cited at Unsolved label and official problem description.Also cited at Editorial research route recorded 2026-07-31.Source used to formulate or check the problem record.Source used to assess the problem's recorded status.Provides the authoritative current status and exact target.Source named by the research packet.
Dave Platt and Tim Trudgian, “The Riemann hypothesis is true up to 3·1012”. Bulletin of the London Mathematical Society 53(3) (2021), 792-797. DOI 10.1112/blms.12460. Abstract and main theorem. ↗journal article · primary source · checked 2026-08-01Source use: original summary.Rigorous finite-height verification and simplicity result.

An original CC0 restatement prepared by TheoremDB maintainers.

[#P31] Riemann hypothesis

1Context

2Definitions

3What counts as a solution

1Status

1Packet records

Recent contributions

Computational notes

2See also

1Lean verification

What the formal work covers

1References

Discussion

Report comment

Background

Definitions

What counts as a solution

What's known

Formal statement

Recorded work

Record & provenance

Flag this problem

Sign in to follow

Report a problem

1Context

2Definitions

3What counts as a solution

1Status

1Packet records

Recent contributions

Computational notes

2See also

1References

Add a comment

Background

Definitions

What counts as a solution

What's known

Formal statement

Recorded work

Related work

Record & provenance