Notes on Cardinal's Matrices

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.