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 generating functions. For more than twenty years progress was made in extending the validity of the conjecture, but recent computer experiments led to counterexamples that show the limits of the original conjecture and pointed the way to a revised conjecture. This paper describes the history of the problem, several numerical examples, and the revised conjecture.