Number-Theoretic Random Walks Project Report

Do numbers play dice? In many ways, the answer seems to be yes. For example, whether an integer has an even or odd number of factors in its prime factorization is a property that behaves much like the outcome of a coin toss: The even/odd sequence obtained, say, from a stretch of one hundred consecutive integers, is, at least on the surface, indistinguishable from the heads/tails sequence obtained by tossing a coin one hundred times. We may thus think of the “even or odd number of prime factors” property as a “digital coin flip”, encoded by a function f(n) that takes on the value +1 if n has an even number of prime factors and −1 if n has an odd number of prime factors.1 Many functions in number theory exhibit such random-like behaviors and can thus serve as a “digital coin flip function”. Our project is part of an ongoing project aimed at exploring the (non)-randomness of such functions geometrically by studying certain “random walks” in the plane formed with these functions. Such random walks provide a natural way to visualize and quantify the degree of randomness of these sequences. They can help detect and explain hidden patterns, but can also reveal new phenomena that have yet to be explained.