Zeta explicit formula zero free error balance

Given: The data and hypotheses of the statement.

The truncated von Mangoldt explicit formula: For $x,T\ge 2$, ${\psi }_0(x)-x=-{\sum }_{\mathrm{\mid }\mathrm{\Im }\rho \mathrm{\mid }<T}\frac{x^{\rho }}{\rho }-\frac{{\zeta }^{\mathrm{\prime }}(0)}{\zeta (0)}-\frac{1}{2}\log (1-x^{-2})+O\text{\hspace{0.17em}⁣}(\frac{x{\log }^2(xT)}{T}+(\log x)\min \{1,\frac{x}{T⟨x⟩}\}),$ where $⟨x⟩$ is the distance to the nearest prime power other than possibly $x$. The zero sum is finite and counts multiplicities.

Riemann zeta classical zero free region: There is an absolute $c_0>0$ such that $\zeta$ has no zeros in $\sigma \ge 1-c_0\mathrm{/}\log (\mathrm{\mid }t\mathrm{\mid }+2)$. The pole at $s=1$ is not a zero.

Zeta reciprocal zero sum bound: For $T\ge 2$, the sum of $1\mathrm{/}\mathrm{\mid }\rho \mathrm{\mid }$ over nontrivial zeros with $0<\mathrm{\mid }\mathrm{Im}\rho \mathrm{\mid }\le T$ is $O({\log }^{2}T)$, with multiplicity. Adjoining any real nontrivial zeros preserves the estimate.

The half-weighted Chebyshev function: For $x>0$, define ${\psi }_0(x)={\sum }_{n<x}\mathrm{\Lambda }(n)+\{\begin{array}{ll}\mathrm{\Lambda }(x)\mathrm{/}2,& x\in {\mathbb{Z}}_{>0},\\ 0,& x\mathrm{\notin }{\mathbb{Z}}_{>0}\mathrm{.}\end{array}$ This differs at prime powers from the right-continuous $\psi (x)={\sum }_{n\le x}\mathrm{\Lambda }(n)$ of def-chebyshev-psi-function.