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Approximating the Chebyshev function

The Chebyshev function

f(x) = ∑pk≤x log(p) 

satisfies the Riemann-Mangoldt formula

f(x) = x - ∑w x2/w  - log(2π) - log(1-1/x2)/2 

where the first sum is over the nontrivial roots of the zeta function, where log(2 pi)=zeta'(0)/zeta(0) comes from the pole and log(1-1/x²)/2 is the contribution from the nontrivial zeros.