Mobius Function Calculator

Q: What is the Mertens conjecture?

It claimed |M(x)| < √x always. Odlyzko and te Riele disproved it in 1985, though no explicit counterexample is known.

Work out mobius function instantly with clear inputs, formula shown and shareable results.

About Mobius Function Calculator

The Möbius function is 0 when n has a squared prime factor, and otherwise (−1)^k where k is the number of distinct primes. It is the key to Möbius inversion and appears throughout analytic number theory.

Formula

Möbius function

μ(1) = 1; μ(n) = (−1)^k if n is a product of k distinct primes; μ(n) = 0 otherwise

Worked example

With the default values below, Mobius Function Calculator returns μ(n) of -1. Change any input above and every figure here is recalculated in your browser.

Inputs used

Numberminimum 1: 30
Also sum μ(k) up toaccepts 1–100,000: 20

Results

μ(n): -1
Meaning: Minus one: squarefree with an odd number of prime factors
Mertens function M(x) = Σ μ(k): -3
Squarefree: Yes, squarefree

Step by step

1Values used
2Möbius function
3μ(n)= -1
4Meaning= Minus one: squarefree with an odd number of prime factors
5Mertens function M(x) = Σ μ(k)= -3
6Squarefree= Yes, squarefree

Assumptions and limits

The result is derived only from the 2 inputs listed above. Anything not among them is not part of the calculation.
In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.
Figures are rounded for display to whole numbers, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.

How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.

Frequently Asked Questions

What is μ(30)?

30 = 2 × 3 × 5 has three distinct primes and no repeats, so μ = −1.

What is the Mertens conjecture?