Notes on Cardinal's Matrices

TL;DR

This paper explores the connection between matrix norm bounds and the Riemann hypothesis, demonstrating equivalences and providing new norm bounds on matrices related to the Mertens function.

Contribution

It introduces a new matrix norm and establishes an equivalence between the Riemann hypothesis and norm bounds in this norm, also providing a deformed matrix version with unconditional bounds.

Findings

Norm bounds on matrices are equivalent to the Riemann hypothesis.

A new matrix norm is proposed for analyzing these matrices.

A deformed matrix version satisfies unconditional norm bounds similar to the Riemann hypothesis.

Abstract

These notes are motivated by the work of Jean-Paul Cardinal on symmetric matrices related to the Mertens function. He showed that certain norm bounds on his matrices implied the Riemann hypothesis. Using a different matrix norm we show an equivalence of the Riemann hypothesis to suitable norm bounds on his matrices in the new norm. Then we specify a deformed version of his Mertens function matrices that unconditionally satisfies a norm bound that is of the same strength as his Riemann hypothesis bound.