  Let $S(x)$ be the number of $n \leq x$ for which a Hadamard matrix of order
$n$ exists. Hadamard's conjecture states that $S(x)$ is about $x/4$. From
Paley's constructions of Hadamard matrices, we have that \[ S(x) =
\Omega(x/\log x). \] In a recent paper, the first author suggested that
counting the products of orders of Paley matrices would result in a greater
density. In this paper we use results of Kevin Ford to show that it does:
\begin{equation}\label{eq:abs} S(x) \geq x/\log x \exp((C+o(1))(\log \log \log
x)^2)\,, \nonumber \end{equation} where $C=0.8178...$.
  This bound is surprisingly hard to improve upon. We show that taking into
account all the other major known construction methods for Hadamard matrices
does not shift the bound. Our arguments use the notion of a (multiplicative)
monoid of natural numbers. We prove some initial results concerning these
objects. Our techniques may be useful when assessing the status of other
existence questions in design theory.
