The Explicit Formula in simple terms

Details The Explicit Formula in simple terms The Explicit Formula in simple terms

1998-10-29

arxiv.org/abs/math/9810169v2

Mathematics

Number Theory

25 pages, plain TeX. The section on the conductor operator has been rewritten, a few final comments added and the references u

Scientific paper

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.

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