The Explicit Formula in simple terms

Abstract

This is a semi-expository paper on the easier aspects of the Explicit Formula for the Riemann Zeta Function. The topics reviewed here include: Weil's criterion for the Riemann Hypothesis and its probabilistic interpretation, various formulations of the contribution corresponding to the real place, Haran's version of the Explicit Formula, and the author's own derivation which puts all places on the same footing. This derivation, an addendum to Tate's Thesis, is in the spirit of Weil's insights towards an adelic understanding of the Explicit Formula. Whereas the analyst would likely formulate the Explicit Formula as a multiplicative convolution, Haran's theorem shows it is also an additive convolution. An intriguing conductor operator is considered whose spectral analysis is equivalent to the Explicit Formula. At a finite place it has a positive cuspidal spectrum. I also comment on the flexibility still left in the Explicit Formula, which shows that it has a symmetry group containing the non-zero rational numbers.