An evil number is a positive integer $n$ that has an even number of $1$s in its binary expansion. Many theorems exist about evil numbers, the most known ones are probably those that involve the Thue-Morse sequence.
However, I find no information about prime numbers having an even number of $1$s in their binary expansion. What is known about such numbers?
While it is obvious that the asymptotic density of evil numbers is $1/2$, is there an equivalent result/conjecture concerning evil primes?
Finally, is there anything known about the sum of the reciprocals of evil primes? (For evil numbers see here.)