I am trying to understand the statement of the Riemann hypothesis (RH) in terms of the determinant of the Redheffer matrices $R_n$.
As outlined in the wikipedia article, the RH is equivalent to a bound on the growth of the determinant of these matrices as a function of $n$. Specifically, the RH is equivalent to the statement that $\operatorname{det}R_n = O(n^{\frac{1}{2}+\epsilon})$ for every positive $\epsilon$.
I take this to mean that for every positive $\epsilon$ there is a real number $A$ and an integer $n_0$ such that
$|\operatorname{det}R_n| \leq A|n^{\frac{1}{2} + \epsilon}|$ whenever $n \geq n_0$.
I do not know how such a bound can be shown to exist for any matrix - must it truly exist for any positive $\epsilon$? I suspect that a concrete, constructive example of a non-trivial infinite dimensional matrix with a similar bound would be very helpful. Any comments on relations between this and Hadamard’s maximal determinant problem would also be appreciated.
I am aware of the generalisation of the notion of a determinant to infinite dimensional vector spaces that occur in the Fredholm theory of trace class operators, but I am not sure how, if at all, this is related to my question.