bibkey: connes1999trace authors: Alain Connes year: 1999 title: Trace formula in noncommutative geometry and the zeros of the Riemann zeta function doi: 10.1007/s000290050042 url: https://arxiv.org/abs/math/9811068v1 claim: The semilocal arithmetic trace formula has a rapidly decreasing remainder for each fixed finite place set and fixed compact Schwartz test. The estimate supplies neither a favorable signed renormalized-trace comparison nor uniform constants for growing support and place sets. strata_touched: [] license: citation-only triage: anchor

Semilocal trace remainder and the remaining sign comparison

The primary text inspected is arXiv:math/9811068v1, 10 November 1998, associated with Selecta Mathematica 5 (1999), 29–106, DOI 10.1007/s000290050042. Its 88-page PDF SHA-256 is dfd4e9924d8980f82e3da11fdea861d318fda8f5c7ba57ee659baf8631975053. The locators below refer to this manuscript, without asserting byte equality with the publisher edition. Section VII’s setup, Theorem 4, equations (33)–(35), Lemma 2 and its final estimate were checked in the primary text. This is source reuse, not independent verification of the entire trace-formula proof or a Lean implementation.

The fixed-test source theorem

Let $k$ be a global field and $S$ a finite set of places containing the infinite places. Section VII uses

$$A_S=\prod _{v\in S}{k}_v,J_S=\prod _{v\in S}{k}_v^{\times },C_S=J_S/O_S^{\times },$$

with the source’s basic character and Haar normalizations. The cutoff projections are $P_{\Lambda }$ and ${\hat{P}}_{\Lambda }=\mathcal{F}{P}_{\Lambda }{\mathcal{F}}^{-1}$, and $R_{\Lambda }={\hat{P}}_{\Lambda }{P}_{\Lambda }$.

For a fixed compactly supported $h\in \mathcal{S}(C_S)$, Theorem 4, manuscript p.31, states

$$\mathrm{Tr}(R_{\Lambda }U(h))=2h(1){\log }^{\prime }\Lambda +\sum _{v\in S}{\int }_{k_v^{\times }}^{\prime }\frac{h(u^{-1})}{|1-u|}{d}^{\ast }u+o(1)(\Lambda \to \infty ).$$

The principal-value normalization is part of the theorem. Here $2{\log }^{\prime }\Lambda$ is the source’s integral of multiplicative Haar measure over $\{{\Lambda }^{-1}\le |a|\le \Lambda \}$ in $C_S$. It must not silently be replaced by a differently normalized logarithm.

Lemma 2, manuscript pp.35–37, strengthens the remainder in the proof. Its error terms are

$${\delta }_q(\Lambda )=\int {\hat{g}}_q(u)(\log |u|-2{\log }^{\prime }\Lambda {)}_{+}du,q\in O_S^{\times },$$

where the $g_q$ arise from a fixed smooth compact lift of $h$. The series of absolute values converges geometrically on the finitely generated $S$-unit group and satisfies

$$\sum _{q\in O_S^{\times }}|{\delta }_q(\Lambda )|=O_{S,h,M}({\Lambda }^{-M})\text{for every }M>0.$$

This is a reusable arithmetic trace remainder. The quantifiers fix $S$, the character and $h$ before taking the cutoff limit. No explicit uniform constants for a changing place set or full changing unit-test family are supplied here.

Common-test and cutoff obligations

The project’s additive support parameter is $L$, its multiplicative support is $[{\lambda }^{-1},\lambda ]$ with $\lambda =e^L$, and its prime-power cutoff is $c=e^{2L}$. Its common autocorrelation has multiplicative support in $[c^{-1},c]$. Relating it to the semilocal test requires the source’s scaling-representation normalization: the unnormalized action and its unitary normalization carry different half-density factors. The existing common-test account retains the actual pole, prime-power and Gamma terms; this note does not replace it by a newly assumed trace identification.

The source’s auxiliary trace cutoff $\Lambda$ is distinct from the support parameter $\lambda$. A FIB support schedule does not determine how the fixed-test error constants behave when $S$ or $h$ changes. The rapid remainder cannot simply be assigned to a growing actual-Weil finite-section error.

More directly, the theorem does not supply the favorable sign needed for the complete renormalized comparison. The expression uses a product of cutoff projections and subtracts $2h(1){\log }^{\prime }\Lambda$. A nonnegative unrenormalized trace, even if separately established for a chosen test, does not by itself bound the remainder after that subtraction. The source’s full global discussion retains an RH-strength trace-formula obligation; the fixed finite-$S$ theorem does not discharge it.

Uniformity in every parameter is not logically necessary for every conceivable positivity argument: a valid signed approximation for each fixed test could suffice by taking its pointwise limit. Such a signed comparison has not been obtained here. That is a distinct obligation from the remainder’s rate.

The spectral construction, effective prolate bounds, archimedean signed Sonin comparison and semilocal dual pairing offer reusable inputs with different operator contracts. The archimedean signed comparison has a fixed prime-free support and specified vanishing conditions; its semilocal extension is a separate obligation from this cutoff remainder. None of these notes establishes the missing common-function arithmetic sign or cofinal Weil positivity. RH remains unproved.