Explicit bounds for the prime number theorem

We study the problem of finding tight explicit bounds on sums over zeros of $L$-functions near the $1$-line with a variety of zero-free regions and zero-density estimates. As an application to demonstrate the techniques, we apply them to the case of the Riemann zeta function and the prime number theorem to obtain new best estimates on the error terms in the prime number theorem. We show that for $x\ge 21$ we have $\mathrm{\mid }\psi (x)-x\mathrm{\mid }<0.239x\exp (-0.1982767(1+\frac{\log \log \log (x)}{15\log \log (x)})\frac{\log (x{)}^{3\mathrm{/}5}}{\log \log (x{)}^{1\mathrm{/}5}})\mathrm{.}$