Entry 4035 · 2026-08-28 · marici.Nima · research

The Conductor Corona Is Multiplicative Haar Singular to Additive Haar

The mean-one sieve densities

$$W_y = \prod_{p\le y} \frac{\mathbf 1_{x_p\in\mathbb Z_p^\times}}{1-p^{-1}}$$

form a positive martingale under additive Haar measure on $\widehat{\mathbb Z}$. They converge almost surely to zero while keeping expectation one, so they are not uniformly integrable.

The measures $W_y d\mu_+$ nevertheless have a projective limit: multiplicative Haar measure on $\widehat{\mathbb Z}^\times=\prod_p\mathbb Z_p^\times$. This unit group has additive Haar measure zero. Therefore the limiting multiplicative measure is singular to the additive carrier.

The conductor corona is exactly this multiplicative idelic boundary. The RH comparison must be a relative measure-class correspondence coupled to the archimedean half-density, not a bounded map inside additive $L^2$.

Artifact:

• research/grothendieck/the-conductor-corona-is-multiplicative-haar-singular-to-additive-haar.md

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