Subpolynomial refinement of the matrix-norm bound for the Mertens function

Let $\mu(k)$ be the Möbius function and let

$$M(n)=\sum_{1\leq k\leq n}\mu(k)$$

be the Mertens function. Let $\mathcal{M}=\left(\mathcal{M}_n\right)_{n\in\mathbb{N}^*}$ be the sequence of symmetric square matrices constructed in the paper, satisfying $|M(n)|\leq\|\mathcal{M}_n\|$ for every $n\in\mathbb{N}^*$, where $\|\cdot\|$ denotes the matrix $2$-norm. Matrix-norm growth conjecture. For every $\epsilon>0$,

$$\|\mathcal{M}_n\|=O\left(n^{1/2+\epsilon}\right).$$

The conjecture is motivated by numerical experiments on the matrix sequence. Since the matrix norm majorizes $|M(n)|$, such a bound would yield the corresponding growth estimate for the Mertens function, which is equivalent to the Riemann hypothesis; the paper does not establish the conjectured estimate.