DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-08-31

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The Mertens function $M (x)$

Definition

For a real number $x \geq 1$ , the Mertens function is

M (x) : = \sum_{1 \leq n \leq x} μ (n),

where $μ$ is the number-theoretic Möbius function of The number-theoretic Möbius function $μ (n)$ from prime factorisation.

Remarks

The index condition is $n \leq x$ , so for every integer $m \geq 1$ the function $M (x)$ is constant on $[m, m + 1)$ and jumps only at integers.

Depends on

The number-theoretic Möbius function $\mu(n)$ from prime factorisation

Used by

The Mertens conjecture is false Counterexample

Dependency tree · two levels

7 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

Tom Sanders, Topics in Analytic Number Theory, Chapter 1 (standard reference, not scraped)

The Mertens function M(x)M(x)