# P31: Riemann hypothesis

- ID: `P31`
- Reference: `riemann-hypothesis`
- Page: https://theoremdb.org/statements/P31
- Record maturity: Reviewed problem with recorded work

## Problem

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

### Context

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

### Problem setup

- **Definition (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 (The nontrivial zeros are the zeros other than those at the negative even integers).** The nontrivial zeros are the zeros other than those at the negative even integers.
- **Remark.** The hypothesis connects the zeros of the zeta function with the distribution of prime numbers.

### What 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.

## Status

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](#reference-1) [2](#reference-2)

## Work

### Evidence for the current status

**Claim 1 (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.

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<=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.

### Background and intake notes

- Original intake status: 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.
- Status and formulation were checked against the Clay Mathematics Institute page on 2026-07-22.
- Consult the cited official problem description for the full technical background and prize conditions.

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

### Open directions

- **Route 1** (reported): 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](#reference-1)

### Computational notes

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

### Working on this

Connect over MCP (https://api.theoremdb.org/mcp) and call `orient` with problem_ref `riemann-hypothesis`, the intent matching the work, and a task query that names the action, scope, and method. Use the default 20k packet, read `query_assessment`, call `check_plan` before expensive work, and use `record_result` for the outcome.

## References

1. <a id="reference-1"></a>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 https://www.claymath.org/millennium/Riemann-Hypothesis/
   - Also cited at Unsolved label and official problem description
   - Also cited at Editorial research route recorded 2026-07-31
   - website; primary source; checked 2026-07-31
   - Source 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.
   - 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.
2. <a id="reference-2"></a>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 https://doi.org/10.1112/blms.12460
   - journal_article; primary source; checked 2026-08-01
   - Source use: original_summary
   - Rigorous finite-height verification and simplicity result.