  A subset of the integers larger than 1 is $primitive$ if no member divides
another. Erdos proved in 1935 that the sum of $1/(a\log a)$ for $a$ running
over a primitive set $A$ is universally bounded over all choices for $A$. In
1988 he asked if this universal bound is attained for the set of prime numbers.
In this paper we make some progress on several fronts, and show a connection to
certain prime number "races" such as the race between $\pi(x)$ and li$(x)$.
