George Gasper, Northwestern University, U.S.A.
e-mail: george@math.nwu.edu

Using squares of real-valued functions to prove that the $\Xi^*(z)$ function and certain other entire functions have only real zeros

Abstract

It is shown how squares of real-valued functions can be used to give new proofs of the reality of the zeros of

$$ \Xi^*(z) = 4\pi^2 \int_0^\infty \cosh{9\over 2}u \ e^{-2\pi \cosh 2u} \cos zu \ du,$$

$$  K_{iz}(a) = \int_0^\infty  e^{-a \cosh u} \cos zu \  du, \ \ a > 0,$$

and of some other entire functions.

See latex, dvi, ps and pdf file of the abstract.