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We define two random variables $X_n,Y_n $ on the sample space $\{1,2,3,\cdots,n\}$ with counting measure. We denote by $C_n$ the covariance of theses two random variables: $C_n=Cov(X_n,Y_n)$. What is the assymptotic behavior of $C_n$ at infinity?

$X_n(K)= \text{The number of prime numbers}\qquad p\qquad \text{with} \qquad p\leq k$

$Y_n(K)= \text{The number of prime numbers}\qquad p\qquad \text{with} \qquad p > k$

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    $\begingroup$ Let $\pi$ be the prime counting function. Let $\Omega := \{1,\ldots,n\}$. I think that your random variables are $X_n(k) = \pi(k)$ and $Y_n(k) = \pi(n) - \pi(k)$ (your notation is not clear). Then $C_ n = \mathbb{E}(X_nY_n) - \mathbb{E}(X_n) \mathbb{E}(Y_n) = -\mathbb{E}X_n^2 + (\mathbb{E}X_n)^2$. With the prime number theorem this can be approximated. $\endgroup$
    – Dieter Kadelka
    Commented Jan 1, 2022 at 11:05

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