Randomness in Number Theory
Peter Sarnak · 2012
Number theoretic dichotomy: Either there is a rigid structure (e.g. a simple closed formula) in a given problem, or the answer is difficult to determine and in that case it is random according to some probabilistic law. • The probabilistic law can be quite unexpected and telling. • Establishing the law can be very difficult and is often the central issue. The randomness principle has implications in both directions. ⇒ Understanding and proving the law allows for a complete understanding of a phenomenon. ⇐ The fact that a very explicit arithmetical problem behaves randomly is of great practical value.