On the de Bruijn-Newman constant: a new approach

Abstract

The conjecture of Newman, proposed in 1976 by Newman, states that all zeros of ${\Xi }_{\aleph }\left(\lambda \right)$ are real for $\aleph \in \mathbb{R}$. Its equivalent statement is that ${\mathbb{M}}_{\aleph }\left(\tau \right)$ has purely imaginary zeros for $\aleph \in \mathbb{R}$. It is well known that ${\mathbb{M}}_{\aleph }\left(\tau \right)$ is an even entire function of order one. This article addresses the product representation for ${\mathbb{M}}_{\aleph }\left(\tau \right)$ by the works of Hadamard and Csordas, Norfolk and Varga. We establish a new class of ${\mathbb{M}}_{\aleph }\left(\tau \right)$ by its series and product. Based on the obtained result, we prove that it has only purely imaginary zeros for $\aleph \in \mathbb{R}$. This implies that the conjecture of Newman is true.

10 pages. Keywords: entire function, even entire function, de Bruijn-Newman constant, conjecture of Newman