Primes, Quantum Chaos, and Computers.

There is a long history of close connections between number theory and numerical analysis. This paper illustrates some of the connections involving the Riemann zeta function. Numerical data, collected as a result of improvements in both hardware and in algorithms, is yielding new information about some of the basic conjectures of number theory. It is also used to explore some new connections that have been proposed between distribution of primes and new areas of physics. This expository paper is based on the author's lecture at the `89 Science and Technology Week Symposium on Number Theory and its Applications.