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Write $\psi (x)={\sum }_{n\le x}\Lambda (n)$, where $\Lambda$ is the Von Mangoldt function. The Riemann–von Mangoldt explicit formula, in its symmetric limiting form for $x>1$, isHere nontrivial zeros are counted with multiplicity, the limit is taken symmetrically through admissible heights, and ${\psi }_0$ assigns half weight at a jump. Its difference from $\psi$ is at most $(\log x)/2$. The Euler product and part (a) exclude zeros with real part at least one. The Functional equation of the Riemann zeta function leaves only the trivial zeros of the Riemann zeta function at negative even integers outside $0<\Re \rho <1$; their already displayed logarithmic correction is $O(x^{-2})$ for large $x$. These terms and the constant are negligible in the requested asymptotic error, rather than literally absent from the exact formula.

A useful truncated explicit formula for the second Chebyshev function is, uniformly for $2\le T\le x$,One may first take a height in $[T,T+1]$ separated from zeros and then adjust to $T$ using the local count. The local zero count for the Riemann zeta function isIt includes multiplicity and is uniform in real $t$. It follows by subtracting the Riemann–von Mangoldt formula at endpoints, handling bounded heights separately and using conjugation for negative heights. Both closed endpoints change the count only by another local bound.

By part (a), every nontrivial zero $\rho =\beta +i\gamma$ with $|\gamma |\le T$ satisfies $\beta \le 1-c_1/{\log }^{9}T$ for large $T$, after reducing the positive constant. The local zero count for the Riemann zeta function givesThere are finitely many zeros at bounded height, none at zero or at one, so that part of the sum is bounded. The truncated explicit formula for the second Chebyshev function now yieldsBalance the exponent losses $c_1\log x/(\log T{)}^9$ and $\log T$ by choosing $\log T=(c_1\log x{)}^{1/10}$. This is the optimal order obtainable from these two errors: making either exponent larger forces the other smaller. Thus, for a positive constant $c_2$,The logarithmic prefactor can be absorbed by reducing $c_2$. This is the prime number theorem error from a logarithmic zero-free region with ninth-power width.

Under the Riemann hypothesis, $|x^{\rho }|=x^{1/2}$. The same reciprocal-zero sum bounds the zero contribution by $O(x^{1/2}{\log }^{2}T)$. Taking $T=x$ makes the truncation error $O({\log }^{2}x)$, so

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 $x{\log }^{2}x\exp (-c\log x/(\log T{)}^A)$ and $x{\log }^{2}x/T$. 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.

For $2\le T\le x$, the truncated Perron formula and a contour shift give the displayed form of the Riemann–von Mangoldt explicit formula. The pole at one contributes $x$ and the Nontrivial zeros of the Riemann zeta function contribute the finite sum. One can choose a separated height in $[T,T+1]$ and then restore $T$ using the local zero count for the Riemann zeta function. The error absorbs the half-weight discrepancy at a prime power. Outside this range of $T$, one needs the fuller error with the near-integer terms retained.