  We discuss the application of the determinantal method to the proof of the
Riemann hypothesis. We start from the fact that, if a certain doubly infinite
set of determinants are all positive, then the hypothesis is true. This
approach extends the work of Csordas, Norfolk and Varga in 1986, and makes
extensive use of the results described by Karlin. We have discovered and proved
or conjectured relations in five areas of the problem, as summarized in the
Introduction. Further effort could well lead to more progress.
