Alphabeta Math
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)∣<xfor 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
1.1

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

given
2.1

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

step 1.1∎

Depends on

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