Primes, Quantum Chaos, and Computers

The primes, which are the fundamental building blocks of the multiplicative structure of the integers, have fascinated mathematicians at least since the time of the ancient Greeks. Euclid already knew that there are infinitely many primes, and his elegant proof is still commonly used today. Further progress in studying primes was made over the succeeding centuries. However, many very basic questions remain unanswered, such as that of accurate determination of the number of primes below a given bound. This paper describes some of the computational work (both actual calculations and algorithmic improvements) that has been done in this area. It is an interesting story that involves many interactions over several centuries between number theory and numerical analysis, and in recent years has also included possible new connections to physics.