It is conjectured that Hadamard matrices exist for all orders 4t (t>0). However, despite a sustained effort over more than five decades, the strongest overall existence results are asymptotic results of the form: for all odd natural numbers k, there is a Hadamard matrix of order k2[a+blog2k]
Nearby in the stack
, where
a
and
b
are fixed non-negative constants. To prove the Hadamard Conjecture, it is sufficient to show that we may take
a=2
and
b=0
. Since Seberry's ground-breaking result, which showed that we may take
a=0
and
b=2
, there have been several improvements where
b
has been by stages reduced to 3/8. In this paper, we show that for all
ε>0
, the set of odd numbers
k
for which there is a Hadamard matrix of order
k22+[εlog2k]
has positive density in the set of natural numbers. The proof adapts a number-theoretic argument of Erdos and Odlyzko to show that there are enough Paley Hadamard matrices to give the result.