The Möbius Landscape

The Möbius function μ(n) through the lens of Matula trees. μ(n) = 0 precisely when the tree has duplicate subtrees — when n has a squared prime factor. The Mertens function M(n) = Σμ(k) accumulates this arithmetic terrain.

Range: –

μ(n) = +1 (even # distinct primes)

μ(n) = −1 (odd # distinct primes)

μ(n) = 0 (has squared factor)

1 M(n) — Mertens function 200

μ(n) values — hover for details

Hover over chart

Select a number from the chart above

Hover or click on any point in the Mertens chart or μ strip to see the Matula tree for that number, its factorization, and why μ has that value. The tree-theoretic interpretation: μ(n) = 0 precisely when the Matula tree A(n) has duplicate subtrees at some vertex — that is, when the number has a repeated prime factor.