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A surprising generalization of the Möbius function
Abstract
We introduce a broad generalization of a recursive formula for the Möbius function μ(n) = 1−n−∑d=2n−1μ(d)[n/d] due to George Spencer-Brown. By replacing the greatest integer function [n/d] in this classical recurrence with an arbitrary arithmetic function f([n/d]) with f(1)=1, we define a new generalized family of functions, denoted ⋆(n). We prove that ⋆(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.
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BibTeX
@misc{andrews2026surprising,
title = {{A surprising generalization of the Möbius function}},
author = {George E. Andrews and Louis H. Kauffman and Divyamaan Sahoo},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.25628},
eprint = {2609.25628},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.25628},
}
RIS
TY - UNPB
TI - A surprising generalization of the Möbius function
AU - George E. Andrews
AU - Louis H. Kauffman
AU - Divyamaan Sahoo
PY - 2026
DA - 2026/09/22
PB - arXiv
DO - 10.48550/arxiv.2609.25628
UR - https://arxiv.org/abs/2609.25628
AB - We introduce a broad generalization of a recursive formula for the Möbius function μ(n) = 1−n−∑_d=2^n−1μ(d)[n/d] due to George Spencer-Brown. By replacing the greatest integer function [n/d] in this classical recurrence with an arbitrary arithmetic function f([n/d]) with f(1)=1, we define a new generalized family of functions, denoted ⋆(n). We prove that ⋆(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.
ER -