In this paper we study quota complexes (or equivalently in the case of scalar weights, threshold complexes) and 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. | Turkish Journal of Mathematics Research Article Turk J Math (2013) 37: 511 – 539 ¨ ITAK ˙ c TUB doi: Threshold complexes and connections to number theory 1 Jonathan PAKIANATHAN,1 Troy WINFREE2,∗ Department of Mathematics, University of Rochester, Rochester, NY 14627 . 2 Seattle, WA 98107 . Received: • Accepted: • Published Online: • Printed: Abstract: In this paper we study quota complexes (or equivalently in the case of scalar weights, threshold complexes) and 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 frames 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 existence of odd perfect numbers among other things. This builds on the original work of A. Bj¨ orner who had studied similar topological formulations for the Riemann Hypothesis and prime number theorem. 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. Key words: Quota system, threshold complex, persistent homology, Goldbach conjecture, Riemann Hypothesis, random complexes 1. Introduction In the area of voting theory in political science, monotone yes/no voting systems are studied. If V is a set of voters, then a subset of voters, C ⊂ V , is called a losing coalition if the voters in C are not sufficient to force an initiative to pass. One can form a simplicial complex
