Complex Zero Decreasing Sequences and the Riemann Hypothesis II

Abstract

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 $$T = \left\{ \gamma \right\}_{k = 0}^\infty $$ such that for any real polynomial $$p\left( x \right): = \sum\nolimits_0^n {{a_k}{x^k}} $$ , the polynomial $$\sum\nolimits_0^n {{\gamma _k}{a_k}{x^k}} $$ 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 ξ-function, some numerical results will highlight a quantitative version of the dictum that “the Riemann Hypothesis, if true, is only barely so”.