Purpose

The Möbius inversion formula allows recovery of an arithmetic function $g$ from its summatory version $f$ defined by $f\left(n\right)={\sum }_{\mathrm{d|n}}^{}g\left(d\right)$. Given $f$ for $n$ up to some limit, the inversion states $g\left(n\right)={\sum }_{\mathrm{d|n}}^{}\mu \left(\frac{n}{d}\right)f\left(d\right)$ where $\mu$ is the Möbius function.

The Möbius function is defined as $\mu \left(n\right)=\left\{\begin{array}{cc}1& \text{if}n=1\\ 0& \text{if}n\text{has a squared prime factor}\\ \mathrm{-1}& \text{if}n\text{is squarefree with an odd number of prime factors}\end{array}\right\}$. Its multiplicative property makes Möbius inversion a key result in multiplicative number theory.

When you supply a list of values for $f(n)$, the calculator applies the inversion formula to compute $g(n)$ for each integer up to your list length. Choosing the g(n) option performs the forward Dirichlet convolution instead, summing your input across the divisors of each n to recover $f(n)$. This two-way capability makes it easy to study how accumulative and pointwise functions relate.

One classic application is recovering an arithmetic function from its Dirichlet convolution with the constant 1 function. For example, Euler’s totient function $\phi (n)$ can be expressed as a sum over divisors of $n$. Using Möbius inversion on that sum reveals an explicit formula for $\phi (n)$ in terms of the Möbius function. Similar reasoning allows you to derive identities for divisor functions and other important sequences.

The tool helps illuminate the structure of multiplicative functions by letting you test how changing $f(n)$ affects the resulting $g(n)$. Try inputting values from well-known sequences, such as the sum of divisors or the identity function, and observe how the inversion isolates individual divisor contributions. Because all calculations happen in your browser, you can explore different cases rapidly without specialized software.

Beyond recreational number theory, Möbius inversion appears in analytic problems where sums over divisors arise naturally. It underlies proofs in the study of the Riemann zeta function and Dirichlet series, offering a method to peel apart convolutions of arithmetic functions. Exploring small examples with this calculator can provide intuition before diving into those deeper applications.

Even if you are new to number theory, experimenting with small sequences can reveal fascinating patterns. Start with simple functions like the identity or constant one and observe how the inversion redistributes weight across divisors. You might notice that squarefree numbers behave differently from those with repeated prime factors—a direct consequence of the Möbius function’s definition.

Once you grasp the basics, consider exploring connections to other combinatorial identities. Möbius inversion has analogues in partially ordered sets and incidence algebras, where it helps invert relations defined by inclusion. The techniques used here mirror those ideas in a more accessible setting of integers and divisors, providing a bridge to broader mathematical concepts.

For deeper experimentation, try computing both directions on the same sequence. Input a short g(n), derive its f(n), then feed those results back in with the opposite mode to verify that you recover the original numbers. Small discrepancies may arise from rounding, underscoring the value of exact arithmetic in theoretical work.

The underlying algorithm iterates over every divisor of each n, so runtime grows with both sequence length and the average number of divisors. It is quick for small inputs but can slow noticeably for long lists. Advanced implementations cache Möbius values or use convolution techniques to accelerate large-scale problems.

Beyond number theory, similar inversion ideas appear in signal processing and combinatorics. Recognizing the pattern in this discrete case builds intuition for transforms like the z-transform or Möbius inversion on partially ordered sets. Exploring sequences here therefore lays groundwork for more sophisticated mathematical tools.

Because the method alternates addition and subtraction based on divisor parity, noisy or approximate data can produce exaggerated swings. If you are analyzing experimental measurements, consider rounding inputs carefully or using rational numbers to maintain stability.

The copy button below the results lets you paste computed sequences into notes, research drafts, or online discussions. Recording interesting examples makes it easier to spot patterns or share insights with fellow enthusiasts.