A surprising generalization of the Möbius function

Abstract

We introduce a broad generalization of a recursive formula for the Möbius function $\mu (n)=1-n-{\sum }_{d=2}^{n-1}\mu (d)\left[\frac{n}{d}\right]$ due to George Spencer-Brown. By replacing the greatest integer function $\left[\frac{n}{d}\right]$ in this classical recurrence with an arbitrary arithmetic function $f\left(\left[\frac{n}{d}\right]\right)$ with $f(1)=1$, we define a new generalized family of functions, denoted $\star (n)$. We prove that $\star (p)=-1$ if and only if $p$ is prime. This result yields a surprising algebraic characterization of primes and reveals a deep structural property underlying divisor sums and the greatest integer function.