How to understand the explicit formula for zeta function?

Question

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?

Answer 1

The analogy between zeta functions and trace formulas goes at least to Selberg, when he proved his famous trace formula for hyperbolic surfaces and the result turned out to resemble Weil's generalization of Riemann's explicit formula quite a bit.

Riemann-Weil explicit formula:

\begin{equation*} \begin{split} \sum_\gamma h(\gamma)=\frac{1}{2\pi}\int_{-\infty}^{+\infty} h(r) \frac{\Gamma'}{\Gamma}\left(\frac{1}{4}+\frac{1}{2}ir \right)dr &+h\left(\frac{i}{2}\right)+h\left(-\frac{i}{2}\right)\\ &-g(0)\ln\pi-2\sum_{n=1}^\infty\frac{\Lambda(n)}{\sqrt{n}}g(\ln n) \end{split} \end{equation*} Selberg trace formula:

\begin{equation*} \begin{split} \sum_{n=0}^\infty h(r_n)=\frac{\mu(F)}{4\pi}\int_{-\infty}^{+\infty} rh(r) \tanh (\pi r)dr &+\sum_n \Lambda(n) g(\ln N(n)) \end{split} \end{equation*}

There's a big literature on this type of question, and the interplay of number theory, spectral analysis, mathematical physics... I recomend section 3 of Lagarias' survey "The Riemann Hypothesis: Arithmetic and Geometry" for references.

Connes approach started with

• "Formule de trace en géométrie non-commutative et hypothèse de Riemann" (1996)

And was completed (as far as I know) in

• "Trace formula in noncommutative geometry and the zeros of the Riemann zeta function" (1998)

The main result says that given a global field $K$ and a character $\alpha=\prod_v \alpha$ of the space of adele classes $A/K$, and any adecuate test function $h$, we have:

$$\underbrace{\widehat{h}(0)+\widehat{h}(1)-\sum \widehat{h}(\mathcal{X},\rho)}_{\text{spectral side}}=\underbrace{\sum_v \int_{K_v^*}' \frac{h(u^{-1})}{|1-u|}d^*u}_{\text{arithmetic side}}$$