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Proving Polynomial Riemann Hypotheses through Eigenfunctions

The papers, [D. Bump, K-K. Choi, P. Kurlberg and J. Vaaler (2000). A Local Riemann Hypothesis, I. Math Zeitschrift , 233 , 1-18. C. Ryavec and Ö. E g ecio g lu (1999). A Polynomial Riemann Hypothesis, in preparation.], show that certain classes of polynomials satisfy Riemann hypotheses through a variety of arguments. In this paper, we will focus on one of the techniques described in these papers, which is based on the fact that polynomial eigenfunctions of a certain class of operators satisfy a Riemann hypothesis. We show that this approach cannot be used for arbitrary polynomials satisfying the Riemann hypothesis and characterize quartic polynomial eigenfunctions.