Zeros of smallest modulus of functions resembling exp(z)

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.

Keywords

• Bessel function
• characteristic function
• exponential function
• exponential polynomial
• Graeffe's method
• Laplace transform
• multiply positive function
• normal density
• probability density function
• variance.