I am interested in the distribution of the parity of $\pi(x)$, the prime counting function, over the natural numbers.


Let:
  

$\ \ E\_n := \left\{ k \in \left\{1,\dots,n\right\} : \pi(k) \equiv 0 \mod 2 \right\}\ \ $ and $\ \ O\_n :=\left\{ k \in \left\{1,\ldots,n\right\} : \pi(k) \equiv 1 \mod 2 \right\}$. 


My question: is it true that


$$\lim\_{n \to \infty} \frac{|E\_n|}{n} = \lim\_{n \to \infty} \frac{|O\_n|}{n} = \frac{1}{2}$$
  

Are any methods known for calculating or estimating these limits?



Parity of the Prime Counting Function
The limits you conjecture are natural, and they are currently open. I believe the best known result is by Ping Ngai Chung and Shiyu Li, who proved that
$$
 \liminf\_{n \to \infty} \frac{|E\_{n}|}{n}
$$
and
$$
 \liminf\_{n \to \infty} \frac{|O\_{n}|}{n}
$$
are both $\geq \frac{1}{64}$. They can improve this to $\frac{1}{8}$ if they assume the Hardy-Littlewood prime $k$-tuples conjecture. The main tool used is Selberg's sieve and their paper appears in Integers in 2013. (See <http://www.integers-ejcnt.org/vol13.html> - the paper is A79.) They mention their result was obtained almost simultaneously by M. Alboiu.


