Truncated explicit formula for the second Chebyshev function ($\psi (x)=x-{\sum }_{|\Im \rho |\le T}{x}^{\rho }/\rho +O(x{\log }^2(xT)/T+\log x)$)

For $2\le T\le x$, the truncated Perron formula and a contour shift give the displayed form of the Riemann–von Mangoldt explicit formula. The pole at one contributes $x$ and the Nontrivial zeros of the Riemann zeta function contribute the finite sum. One can choose a separated height in $[T,T+1]$ and then restore $T$ using the local zero count for the Riemann zeta function. The error absorbs the half-weight discrepancy at a prime power. Outside this range of $T$, one needs the fuller error with the near-integer terms retained.