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    Prime Counting Function

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    Limit

    lim_(x->-∞) π(x) = 0

    Alternative representations

    π(x) = sum_(i=1)^n 1 for p_n<=x<p_(1 + n)

    π(x) = sum_(k=1)^floor(x) θ(x - p_k) for (p_k element P and x element R and x>=0)

    π(x) = ( sum_(k=2)^floor(x) floor(ϕ(k)/(-1 + k)) = sum_(k=2)^floor(x) floor(ϕ(k)/(-1 + k)))

    π(x) = - sum_(k=1)^floor(log(2, x)) μ(k) sum_(n=2)^floor(x^(1/k)) floor(x^(1/k)/n) μ(n) Ω(n)

    Series representations

    π(x) = sum_(i=1)^n 1 for p_n<=x<p_(1 + n)

    π(x) = sum_(k=1)^x( piecewise | 1 | k element P 0 | otherwise) for (x element Z and x>0)

    π(x) = -1 + sum_(k=3)^x((-2 + k)! - k floor(((-2 + k)!)/k)) for (x element Z and x>3)

    Integral representation

    π(x) = -1 + x - 1/(2 π) integral_0^(2 π) ( sum_(m=1)^x cos(t product_(k=1)^(-1 + m) product_(j=1)^(-1 + m)(j k - m))) dt for (x element Z and x>0)

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