International Journal of Mathematics and Mathematical Sciences
Volume 5 (1982), Issue 3, Pages 417-439
doi:10.1155/S0161171282000428
Zeros of smallest modulus of functions resembling exp(z)
Department of Mathematics, University of Illinois, Urbana 61801, Illinois, USA
Received 17 July 1981
Copyright © 1982 Kenneth B. Stolarsky. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
To determine (in various senses) the zeros of the Laplace transform of a signed mass distribution is of great importance for many problems in classical analysis and number theory. For example, if the mass consists of finitely many atoms, the transform is an exponential polynomial. This survey studies what is known when the distribution is a probability density function of small variance, and examines in what sense the zeros must have large moduli. In particular, classical results on Bessel function zeros, of Szegö on zeros of partial sums of the exponential, of I. J. Schoenberg on k-times positive functions, and a result stemming from Graeffe's method, are all presented from a unified probabilistic point of view.