Calculator
Example Data Table
| n | Prime Factorization | Square-Free | μ(n) | M(n) |
|---|---|---|---|---|
| 1 | 1 | Yes | 1 | 1 |
| 2 | 2 | Yes | -1 | 0 |
| 4 | 2^2 | No | 0 | -1 |
| 6 | 2 × 3 | Yes | 1 | -1 |
| 10 | 2 × 5 | Yes | 1 | -1 |
| 12 | 2^2 × 3 | No | 0 | -2 |
Formula Used
The Möbius function μ(n) is defined by three rules:
- μ(1) = 1
- μ(n) = 0 if any prime square divides n
- μ(n) = (-1)k when n is square-free with k distinct prime factors
Mertens function: M(n) = ∑ μ(k), for 1 ≤ k ≤ n
Divisor identity: ∑ μ(d) over all divisors d of n equals 1 when n = 1, otherwise 0.
How to Use This Calculator
- Enter the target integer n for the primary Möbius result.
- Set the range start and range end for the table and chart.
- Choose whether you want one result, a full range, or both.
- Select divisor visibility and the chart style you prefer.
- Press Calculate Möbius Function to generate the report.
- Use the CSV or PDF buttons to export the produced results.
Frequently Asked Questions
1. What does the Möbius function measure?
It classifies an integer by its prime factor structure. The value becomes 0 when any prime square divides the number. Otherwise, the sign depends on whether the count of distinct prime factors is even or odd.
2. Why is μ(1) equal to 1?
The value 1 is the multiplicative identity and has no prime factors. Defining μ(1)=1 keeps many number theoretic identities clean, especially inversion formulas and divisor sum relationships.
3. When does μ(n) become zero?
It becomes 0 whenever n contains a repeated prime factor. Examples include 4, 12, 18, and 50 because each has a squared prime dividing it.
4. What does square-free mean here?
A square-free integer is not divisible by any perfect square greater than 1. Numbers like 6 and 30 are square-free, but 8 and 45 are not.
5. What is the Mertens function?
The Mertens function M(n) is the running total of Möbius values from 1 through n. It helps visualize cancellations that appear in multiplicative number theory.
6. Why show the divisor identity sum?
The divisor sum verifies a classic identity: the sum of μ(d) over all divisors of n is 1 only for n=1, and 0 otherwise.
7. Can I export the generated table?
Yes. The page includes CSV and PDF export buttons. CSV is useful for spreadsheets, while PDF is better for sharing a fixed report snapshot.
8. Why is the visible range limited?
Large tables and dense graphs can slow browsers and clutter interpretation. A tighter range keeps the chart responsive and makes patterns in μ(n) and M(n) easier to inspect.