Smoothed explicit formula for the Chebyshev function (${\Psi }_1(x)=x^2/2-{\sum }_{\rho }{x}^{\rho +1}/(\rho (\rho +1))+O(x)$)

For $x>1$, moving the Mellin contour past the pole at one, the Nontrivial zeros of the Riemann zeta function, and the remaining poles givesZeros are included with multiplicity. The Riemann–von Mangoldt formula makes the zero sum absolutely convergent. If every nontrivial zero has real part below one, division by $x^2$ and dominated convergence give ${\Psi }_1(x)\sim x^2/2$. Monotonicity of $\psi$ then recovers $\psi (x)\sim x$ by upper and lower difference quotients.