Complex zero decreasing sequences and the Riemann Hypothesis II
by
George Csordas
Department of Mathematics, University of Hawaii

A long-standing open problem (the Karlin-Laguerre problem) in the theory of distribution of zeros of real entire functions requires the characterization of all real sequences T={\gamma} _k=0^\infty such that for any real polynomial p(x):=\sum₀ⁿ a_kx^k, the polynomial \sum₀ⁿ \gamma_k a_kx^k has no more nonreal zeros than p(x) has. The sequences T which satisfy the above property are called complex zero decreasing sequences. While the Karlin-Laguerre problem has remained open, recently there has been significant progress made in a series of papers by A. Bakan, T. Craven, A. Golub and G. Csordas. In particular, it follows that under a mild growth restriction, an entire function, f(z), of exponential type has only real zeros, if the sequence T={f(k)}_k=0^\infty is a complex zero decreasing sequence. These results yield new necessary and sufficient conditions for the validity of the Riemann Hypothesis. Applying these conditions to the Riemann \xi-function, some numerical results will highlight a quantitative version of the dictum that ``the Riemann Hypothesis, if true, is only barely so.''

Date received: May 6, 2001

Copyright © 2001 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 # cahk-24.