Inverse Mellin transform contour wrapping non-trivial zeros ρ of the Riemann zeta function
Perron's Formula expresses the step-vector prime counting function $\psi(x) = \sum_{n \le x} \Lambda(n)$ as an inverse Mellin transform contour integral over the complex plane: $\psi_0(x) = \frac{1}{2\pi i} \int_{c - i\infty}^{c + i\infty} -\frac{\zeta'(s)}{\zeta(s)} \frac{x^s}{s} ds$.
By shifting the vertical line of integration $\text{Re}(s) = c > 1$ to the left across the critical strip $0 < \text{Re}(s) < 1$, Cauchy's Residue Theorem captures poles at $s = 1$ (main term $x$) and at non-trivial zeros $s = \rho$ (oscillatory terms $-x^\rho / \rho$), yielding the explicit formula $\psi(x) = x - \sum_\rho \frac{x^\rho}{\rho} - \ln(2\pi)$.
This interactive diagram renders the complex $s$-plane, displaying the vertical integration path $c \pm iT$, the main pole at $s=1$, trivial zeros along the negative real axis, and non-trivial zeros aligned on the critical line $\text{Re}(s) = 1/2$.