Prime number theorem error from a fixed zero-free strip ($\psi (x)=x+O(x^{1-c}{\log }^{2}x)$)

If all nontrivial zeta zeros satisfy $\Re \rho \le 1-c$ with fixed $0<c<1/2$, the Riemann–von Mangoldt formula gives ${\sum }_{|\rho |<T}|\rho {|}^{-1}\ll {\log }^{2}T$. The truncated explicit formula for the second Chebyshev function then bounds the error by $O(x^{1-c}{\log }^{2}T+x{\log }^{2}x/T)$. Choosing $T=x^c$ gives the displayed bound and, for every positive $\epsilon$, $O_{\epsilon }(x^{1-c+\epsilon })$.

// Target: number-theory.bigb