The explicit formula for the zeta function, e.g.

$$\psi_0(x) = \dfrac{1}{2\pi i}\int_{\sigma-i \infty}^{\sigma-i \infty}\left(-\dfrac{\zeta'(s)}{\zeta(s)}\right)\dfrac{x^s}{s}ds=x-\sum_\rho\frac{x^\rho}{\rho} - \log(2\pi) -\dfrac{1}{2}\log(1-x^{-2})$$

(where $\psi_0$ is the normalized Chebyshev function, $\sigma > 1$) relating zeroes of the $\zeta$ function and primes is certainly of much importance.

How does one understand this beyond what's normally given by a standard textbook on analytic number theory?

According to Connes

The noncommutative space of adele classes of a global field provides a framework to interpret the explicit formulas of Riemann-Weil in number theory as a trace formula.

Can someone elaborate on this? How do we interpret this as a trace formula?

youunderstand the explicit formula ? Because what Connes wrote is one of the most abstract (and complicated) way to understand it. $\endgroup$ – reuns Dec 4 '16 at 11:12