On Certain Points Requiring Clarication in the Published Proof of the Non-Negativity of the de Bruijn - Newman Constant

The proof of Rodgers and Tao that the de Bruijn-Newman constant is nonnegative is among the most original and delicate achievements in the modern theoryof zeta zeros. The purpose of the present note is not to dispute that theorem, nor tolessen the depth of the argument, but only to record a few places where the publishedproof appears to call for further clarification or correction. One point concerns adirect discrepancy in the passage from equation (9) to equations (54)-(55). A secondconcerns Proposition (28), where the argument appears to move from fixed boundarycentres to moving boundary centres without an explicit comparison principle. Athird concerns the final contradiction, where a local rigidity statement is pairedwith a small-gap input stated in a more global form. None of these observationsis offered in a destructive spirit. The aim is simply to indicate, as carefully andrespectfully as possible, the places where the printed proof seems to ask for onemore line, one more lemma, or one explicit correction, so that an argument of greatbeauty may also become fully transparent at its most delicate points.