Möbius Transform Calculator

NumbertheoreticdivisorMobiusinversiong(n)=Σd|nμ(d) f(n/d), not complex fractional-linearMobiusmap. Explicit bounded resource/branch and actual observations or independently verified parameters; no unproved infinite assertion certified.

Description

Divisor Möbius inversion recovers a sequence from its divisor sums.

Divisor Möbius inversion recovers a sequence from its divisor sums.^[1]

This tool performs the finite number-theoretic transform on one-based observations rather than a complex fractional-linear map.

When to use Möbius Transform Calculator

This tool performs the finite number-theoretic transform on one-based observations rather than a complex fractional-linear map.

How to use Möbius Transform Calculator and how it works

Prepare sequence at one based integers, using the input definitions and matching units below.
Enter the required fields and run Möbius Transform Calculator.
Read result.divisorMobiusTransform using the units, signs and ordering explained below.
Correct any rejected inputs and rerun; interpret the result within the stated model and its assumptions.

Input field	Meaning	Value type	Requirement
actualSequenceAtOneBasedIntegers	Sequence [f(1),f(2),...] to invert from divisor sums.	array of number	Required; other constraints below

Require a nonempty finite numeric sequence indexed from one. Copy it and, in ascending n order, subtract each already inverted a_n from every proper multiple index within the sequence. Return the divisor Möbius-inverted values in one-based order. No complex fractional-linear transformation or infinite continuation is performed.

g(n)=sum(mu(d)*f(n/d),d divides n)

Apply the input ordering, units and branch conditions stated above.

Output field	Meaning	Value type
result	Object containing the calculated or selected fields below.	object
result.divisorMobiusTransform	Divisor-inverted numeric sequence in original one-based index order.	array of number

json

{
  "input": {
    "actualSequenceAtOneBasedIntegers": [
      1,
      2,
      3,
      4
    ]
  },
  "output": {
    "result": {
      "divisorMobiusTransform": [
        1,
        1,
        2,
        2
      ]
    }
  }
}

Limitations and assumptions

The calculation uses the supplied observations and model inputs. It does not collect measurements, select a reference population or coefficient, or establish that the model applies to the actual case. The stated branch conditions and output units remain part of the result.

All fields require explicit values. Unexpected fields, wrong types, nonfinite numbers and sparse arrays are rejected. Alignment, bounds and supported-token rules are explained above; extremely large inputs or nonfinite results can also fail. Input errors differ from successful false comparisons, unavailable-domain flags and empty match lists.

Alternative or Complementary methods

Use J-Invariant Fourier Coefficient Calculator for a complementary operation: The standard modular j function has integer Fourier coefficients and constant term 744.

Use Ramanujan Tau Approximation for a complementary operation: Ramanujan tau is the integer coefficient sequence of the modular discriminant's q-product.

References

Möbius Transform Calculator — tool.io

Similar or alternative tools

J-Invariant Fourier Coefficient Calculator
The standard modular j function has integer Fourier coefficients and constant term 744.
Ramanujan Tau Approximation
Ramanujan tau is the integer coefficient sequence of the modular discriminant's q-product.