= Prime number theorem error from a logarithmic zero-free region
{title2=$1-\beta\gg(\log|\gamma|)^{-A}\ \Longrightarrow\ \psi(x)-x\ll x\log^2x\,e^{-c(\log x)^{1/(A+1)}}$}
Suppose $A>0$ and every sufficiently high <Nontrivial zero of the Riemann zeta function> satisfies the displayed gap, with bounded heights also separated from one. The <truncated explicit formula for the second Chebyshev function> and the <local zero count for the Riemann zeta function> give errors $x\log^2x\exp(-c\log x/(\log T)^A)$ and $x\log^2x/T$. Balance them by $\log T\asymp(\log x)^{1/(A+1)}$. This explains how the width of a <zero-free region of the Riemann zeta function> determines the exponential scale in the <Prime number theorem> error.
Back to article page
