THE FISHER-HARTWIG CONJECTURE AND TOEPLITZ EIGENVALUES

The conjecture of Fisher and Hartwig, published in 1968, describes the asymptotic expansion of Toeplitz determinants with singular generatin g functions. For more than twenty years progress was made in extending the validity of the conjecture, but recent computer experiments led t o counterexamples that show the limits of the original conjecture and pointed the way to a revised conjecture. This paper describes the hist ory of the problem, several numerical examples, and the revised conjec ture.