Quota Complexes, Persistant Homology and the Goldbach Conjecture

TL;DR

This paper introduces quota complexes to study topological changes related to number theory problems, providing new formulations for conjectures like Goldbach's and the Riemann Hypothesis, and explores their properties through both deterministic and random models.

Contribution

It defines quota complexes and demonstrates their application in formulating and analyzing major number theory conjectures and theorems, linking topology with number theory.

Findings

Quota complexes encode prime distribution and number theory conjectures.

Topological formulas relate to prime number theorem and twin prime conjecture.

Expected topological quantities connect to L-series and Euler products.

Abstract

In this paper we introduce the concept of a quota complex and study how the topology of these quota complexes changes as the quota is changed. This problem is a simple "linear" version of the general question in Morse Theory of how the topology of a space varies with a parameter. We give examples of natural and basic quota complexes where this problem codifies questions about the distribution of primes, squares and divisors in number theory and as an example provide natural topological formulations of the prime number theorem, the twin prime conjecture, Goldbach's conjecture, Lehmer's conjecture, the Riemann Hypothesis and the existance of odd perfect numbers among other things. We also consider random quota complexes associated to sequences of independent random variables and show that various formulas for expected topological quantities give L-series and Euler product analogs of interest. Keywords: Quota system, persistant homology, Goldbach conjecture, Riemann Hypothesis, random complexes.