# Mertens Conjecture

**Type:** Disproven Conjecture  
**Domain:** Named Results / Number Theory  
**Posed:** 1897, Franz Mertens  
**Disproved:** 1985, Andrew Odlyzko and Herman te Riele  
**Codex URL:** /mathematics/named-results/mertens-conjecture/  
**Entry status:** Live — v1.0 (2026-05-27)

## Summary
Conjectured |M(n)|<√n for the Mertens function. Disproven 1985 — without ever finding an explicit counterexample.

## Möbius function and Mertens function
μ(n): 1 if even number of distinct prime factors, −1 if odd, 0 if repeated factor. M(n) = running sum of μ(1)...μ(n).

## The 1985 disproof
Odlyzko and te Riele proved a violation must exist using analytic methods tied to zeta function zeros — without identifying the specific number.

## How large the counterexample likely is
Estimated around n = 10^(10^40) — computationally inaccessible, likely forever.

## Connection to Riemann Hypothesis
Mertens's conjecture (if true) would have implied Riemann Hypothesis. Disproof doesn't resolve Riemann either way — just removes this potential pathway.

## The lesson
A century of numerical support, ultimately overturned — a clean case study in why computational evidence isn't proof.

## Sources
### Tier 1
- Odlyzko, A.M. and te Riele, H.J.J. (1985). J. reine angew. Math.
- Mertens, F. (1897). Sitzungsberichte.
### Tier 2
- Pintz, J. (1987). Astérisque.

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*Mathematics Codex entry v1.0 — added 2026-05-27 — thecodex.expert/mathematics/*