CounterexampleConstruction: Literature-sourcedVerification: Literature-sourcedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-31

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

The Mertens conjecture is false

Statement refuted

The historical statement

$∣ M (x) ∣ < \sqrt{x} for every real x > 1$

is false, where $M$ is the Mertens function of The Mertens function $M (x)$ .

Facts & Assumptions

Given: The historical Mertens conjecture.

Counterexample

technique · direct

1.1given

The cited paper of Odlyzko and te Riele proves that there exists a real number $x > 1$ for which $∣ M (x) ∣ \geq \sqrt{x}$ . So the universal inequality in the Statement refuted fails at that value of $x$ .

2.1step 1.1∎

Hence the Mertens conjecture is false. This example is historical and non-load-bearing for the finite convolution arguments on the A page.

Depends on

The Mertens function $M(x)$

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

2 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

A. M. Odlyzko and H. J. J. te Riele, Disproof of the Mertens conjecture (standard reference, not scraped)