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”.
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Csordas, G. (2003). Complex Zero Decreasing Sequences and the Riemann Hypothesis II. In: Begehr, H.G.W., Gilbert, R.P., Wong, M.W. (eds) Analysis and Applications — ISAAC 2001. International Society for Analysis, Applications and Computation, vol 10. Springer, Boston, MA. https://doi.org/10.1007/978-1-4757-3741-7_9
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DOI: https://doi.org/10.1007/978-1-4757-3741-7_9
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