Zeros and explicit formulas

Let $\psi(x)=\sum_{n\leq x}\Lambda(n)$ for real $x>1$. It jumps at prime powers, including composite prime powers, by the logarithm of their underlying prime. Define $\psi_0(x)$ to use all terms strictly below the endpoint and half the endpoint weight when the endpoint is an integer. Away from prime powers, the two counts agree. This symmetric value is natural for inverse contour integrals, just as symmetric Fourier inversion returns the average of the two one-sided limits at a jump.

For positive real $x$, define $x^\rho$ using the real logarithm, so no branch ambiguity enters. The nontrivial zeros are those in the open critical strip, each repeated according to multiplicity. Sum them with a symmetric height cutoff and let the cutoff tend to infinity through values avoiding zero ordinates. The exact formula includes a constant from the origin and a convergent contribution from the trivial zeros. Omitting these terms would change an exact identity even if they are small relative to a large main term.

On a vertical line with real part greater than one, the logarithmic derivative has the absolutely convergent series with coefficients $\Lambda(n)$. Integrating it against $x^s/s$ and shifting a finite contour left crosses its pole at one and its poles at zeta zeros. Their residues have opposite signs: the main pole contributes $x$, and a zero of multiplicity $m$ contributes $-m x^\rho/\rho$. Horizontal contour pieces cannot be discarded merely because the desired residues have already appeared.

A convenient clean truncated statement restricts $x$ to half-integers with $x\geq5/2$ and $2\leq T\leq x$, with the height not equal to a zero ordinate. In that range the displayed error absorbs the bounded origin and trivial-zero terms. Keeping the endpoint away from every integer controls its distance from prime powers, and the usual contour argument controls the horizontal integrals by choosing nearby safe heights. The full theorem for general endpoints includes a term depending on distance to the nearest distinct prime power; the cleaner formula should not be extended silently beyond its stated range.

Conjugate zeros occur in pairs. Writing one as $\rho=\beta+i\gamma$, its pair contributes twice the real part of $x^\rho/\rho$. Its envelope grows like $x^\beta$ divided by the modulus of the zero, while its phase is proportional to $\gamma\log x$. A larger real part allows a larger correction; a larger height produces faster oscillation in logarithmic position. Because the zero sum is not absolutely convergent, an arbitrary rearrangement is not interchangeable with the prescribed symmetric cutoff.

The prime number theorem is $\psi(x)=x+o(x)$ and follows from appropriate analytic information, including nonvanishing on the line with real part one. Under the additional Riemann-hypothesis assumption, the explicit-formula estimates yield $\psi(x)=x+O(\sqrt{x}\log^2x)$. This is a conditional implication, not a deduction from reflection symmetry alone. In computation, approximating a handful of zeros does not certify either bound: one must also control the remaining zeros, their numerical uncertainty, and the chosen height and endpoint errors.