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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?

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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.

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