= Smoothed explicit formula for the Chebyshev function
{title2=$\Psi_1(x)=x^2/2-\sum_\rho x^{\rho+1}/(\rho(\rho+1))+O(x)$}
For $x>1$, moving the Mellin contour past the <pole> at one, the <Nontrivial zeros of the Riemann zeta function>, and the remaining poles gives
$$
\Psi_1(x)=\frac{x^2}{2}-\sum_\rho\frac{x^{\rho+1}}{\rho(\rho+1)}-x\log(2\pi)+\frac{\zeta'(-1)}{\zeta(-1)}-\sum_{k\ge1}\frac{x^{1-2k}}{2k(2k-1)}.
$$
Zeros 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.
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