= Prime number theorem error from a fixed zero-free strip
{title2=$\psi(x)=x+O(x^{1-c}\log^2x)$}
If all nontrivial zeta zeros satisfy $\Re\rho\leq1-c$ with fixed $0<c<1/2$, the <Riemann–von Mangoldt formula> gives $\sum_{|\rho|<T}|\rho|^{-1}\ll\log^2T$. The <truncated explicit formula for the second Chebyshev function> then bounds the error by $O(x^{1-c}\log^2T+x\log^2x/T)$. Choosing $T=x^c$ gives the displayed bound and, for every positive $\varepsilon$, $O_\varepsilon(x^{1-c+\varepsilon})$.
// Target: number-theory.bigb
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