arXiv subjects

Divyamaan Sahoo

Publications and source records attributed to Divyamaan Sahoo.

3 recordsLinked to original sources

arXiv2026

A surprising generalization of the Möbius function

We introduce a broad generalization of a recursive formula for the Möbius function $μ(n) = 1-n-\sum_{d=2}^{n-1}μ(d)\left[\frac{n}{d}\right]$ 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.

math.NT↗

arXiv2025

Primes Between Squares -- Commentary on Appendix 8 of Laws Of Form

This paper provides a commentary and guide to Appendix 8 of Laws Of Form, which is a chapter (appendix) on number theory in the book Laws of Form by Spencer-Brown. (Spencer-Brown,Laws Of Form,Revised Seventh English edition. Bohmeier Verlag. 2020) This chapter in the book provides Spencer-Brown's proofs of the conjecture that there are at least two prime numbers between any consecutive squared numbers. That there are primes between squares has been a conjecture in number theory since Legendre. In Spencer-Brown's appendix he gives his proofs of the conjecture. Those proofs are a highly original mixture of standard rigorous arguments and also some stated facts about the way numbers behave that would be considered conjectures by most number theorists. These phenomena are very interesting and constitute a deep observation about the nature of number itself. We intend that our guide will enable the reader to gain insight into Spencer-Brown's point of view and that our discussions will be of interest to anyone with curiosity about the theory of numbers.

math.NT↗

arXiv2022

Laws Of Form and the Riemann Hypothesis

This paper is an exposition and review of the research related to the Riemann Hypothesis starting from the work of Riemann and ending with a description of the work of G. Spencer-Brown, culminating in his Denjoy proof of the RH.

math.HO↗