  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.
