The truncated explicit formula has the shape

\begin{equation}\label{1}
\psi(x) =x-\frac{\zeta^{\prime}(0)}{\zeta(0)}-\sum_{|\rho|\leq T}\frac{x^{\rho}}{\rho}+\sum_{n=1}^{\infty}\frac{x^{-2n}}{2n}+O\left(\frac{x\log (Tx)^2}{T} \right) ,
\end{equation}
where $\rho=\beta+it$ are the nontrivial zeros of the zeta function, [1, Explicit Formula], and there are other versions of it.

The final paragraph on the proof of Theorem 6.9 in [2, page 181] states the following:

"... On combining these estimates we conclude that
\begin{equation}\label{2}
\psi(x) =x+O\left(x(\log x^2)\left( \frac{1}{T}+x^{-c/\log T} \right) \right) .
\end{equation}
We choose $T$ so that the two terms in the last factor of the error term are equal, $T = \exp(\sqrt{c\log x})$. With this choice of $T$, the error term above is
\begin{equation}\label{3}
\ll x(\log x)^2\exp(-\sqrt{c\log x})\ll x\exp(-c\sqrt{\log x})
\end{equation}

since we may suppose that $0 < c < 1$. Thus the proof of (6.12) is complete."

**Question.** Why does the proof of Theorem 6.9, based on the zero free region $1-c/\log T$, or any other known zero free region, uses fewer zeros $\rho=\beta+it$ in the approximation to $\psi(x)$. Specifically, $t\leq T< \exp(\sqrt{c\log x})<x$.

We should expect that increasing the number of zeta zeros to $t\leq T=x$, should have a smaller error term $E(x)$ since the approximation to $\psi(x)=\sum_{n\leq x}\Lambda(n)=x+E(x)$ is better. But, it does not, in fact, the error term of the truncated explicit formula (2) exceeds the main term. It has the shape \begin{equation}\label{4} \psi(x)=\sum_{n\leq x}\Lambda(n)=x+O(x(\log x)^2).\end{equation}

Please edit as required to improve the question. Many other paradoxes in the theory of the zeta function are documented in [3].

[1] Harold Davenport, Multiplicative number theory, third ed., Graduate Texts in Mathematics, vol. 74, Springer-Verlag, New York, 2000.

[2] Hugh L. Montgomery and Robert C. Vaughan, Multiplicative number theory. I. Classical theory, Cambridge Studies in Advanced Mathematics, vol. 97, Cambridge University Press, Cambridge, 2007.

[3] David W. Farmer, Currently there are no reasons to doubt the Riemann Hypothesis, arXiv:2211.11671.