# Feedback, Control, and Distribution of Prime Numbers

by Susan Marshall and Donald Smith

Year of Award: 2014

Award: Allendoerfer

Publication Information: Mathematics Magazine, vol. 86, 2013, pp. 189-203.

Summary (adapted from the Prizes and Awards booklet for MathFest 2013)

In this article, the authors describe an unusual application of a technique of mathematical modeling, feedback and control, to a classical mystery of number theory, the distribution of primes. In a famous result due to Gauss, the density of primes is (approximately) inversely proportional to the natural logarithm. The differential equation below reasonably models the density of primes. Here $f(x)$ represents the density of primes:

$f^{\prime}(x) = \frac{f(x) f(\sqrt{x})}{2x}$

Although this is a known application in differential equation literature, it appears to be largely forgotten in number theory. In the process of deriving this model, the authors give the reader a lively introduction to the theory of feedback and control. The authors note that the distribution of prime numbers has an element of randomness, yet it also stays on track, much like a feedback and control system.