On the de Bruijn-Newman constant: a new approach

TL;DR

This paper introduces a new approach to analyze the de Bruijn-Newman constant, proving that a key function has only purely imaginary zeros for real parameters, thereby confirming Newman's conjecture.

Contribution

The paper establishes a new class of functions related to the de Bruijn-Newman constant and proves they have only purely imaginary zeros, confirming Newman's conjecture.

Findings

Confirmed that $ ext{M}_eta( au)$ has only purely imaginary zeros for real $eta$

Provided a new product and series representation for $ ext{M}_eta( au)$

Proved Newman's conjecture using the new class of functions

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.