Is the Riemann Hypothesis Necessary ?
by
Eric Bach
University of Wisconsin

To one unacquainted with number theory, we might explain our interest in the zeroes of the zeta function by saying that we can thereby estimate the error incurred by using logarithm integral to count the primes up to a given bound. As a practical matter, however, we know enough about these zeroes that numerical estimates for prime number sums, accurate enough for most purposes, can be easily obtained. There are other situations in computational number theory, though, where it seems impossible to obtain results that come close to empirical data without assuming the Riemann hypothesis or one of its generalizations. One example is the least witness required to prove a number composite using the strong pseudoprime test, and there are many others. For this reason, Riemann hypotheses have now become a standard tool of the algorithm designer.

This talk will provide an introduction to and survey of these matters.

Date received: April 26, 1999

Copyright © 1999 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cacf-50.