What is known about evil primes? 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.)

There is a very similar Mathoverflow question, in which it was shown that the ratio of odd-bit primes against even-bit primes approaches $1/2$. In fact, the rigourous proof can be found in

C. Mauduit and J. Rivat, _Sur un problème de Gelfond: la somme des chiffres des nombres premiers_ , Ann. Math.