bibkey: hathi2025numberfieldmertens authors: Shehzad Hathi; Ethan S. Lee year: 2025 title: “Mertens’ Third Theorem for Number Fields: A New Proof, Cramér’s Inequality, Oscillations, and Bias” doi: null url: https://arxiv.org/abs/2112.02166v3 claim: The source treats Mertens products over prime-ideal norms, including the golden field; its conditional bounds and bias do not control the ordinary Robin tail, and the ideal observation does not distinguish golden unit iterates. strata_touched: [] license: citation-only triage: anchor

Golden-field Mertens response and the retained arithmetic source

The inspected primary is arXiv:2112.02166v3, revised 6 January 2025, 23 pages, with PDF SHA-256 9342c98d92478a90f0f7128d8ee5915928ca257e3f9d3186ec3a0a42d0d95d7a. Theorems 1 and 3 on p.3, Theorem 5 on p.4, Theorem 22 and Propositions 23–24 on p.17, and equation (31) with its application in §5.4 on pp.18–20 were inspected. This note cites that version; it does not certify the complete proof or a publisher version, rerun its numerical density calculation, or claim Lean verification.

The source’s measure and assumptions

For a number field $K$, the source uses the prime-ideal product

$${\mathcal{P}}_K(x)=\prod _{N\mathfrak{p}\le x}{\left(1-\frac{1}{N\mathfrak{p}}\right)}^{-1},$$

whose Mertens main term is $e^{\gamma }{\kappa }_K\log x$. Here ${\kappa }_K$ is the residue of ${\zeta }_K$ at one, and each prime ideal is counted once. This is distinct from the rational-prime product and from the divisor response $\sigma (N)/N$ at a fixed integer $N$.

The statements have separate hypothesis scopes:

Theorem 1 requires an attained rightmost nonreal zero. Merely excluding a real exceptional zero does not supply that existence condition. Its oscillation conclusion is also not a one-sided bound at every required source. The mean-square and density statements retain their GRH premises and their interval or logarithmic-measure conclusions; neither supplies a bound at a specified Robin-critical cutoff.

The actual golden-field factor is already classical

The project’s golden-ring interface identifies the coordinate ring $\mathbb{Z}[\theta ]$, ${\theta }^2=\theta +1$, with the ring of integers of $K=\mathbb{Q}(\sqrt{5})$. Equation (31), p.18, directly supplies the classical factorization

$${\zeta }_K(s)=\zeta (s)L(s,{\chi }_5),{\chi }_5(n)=\{\begin{array}{ll}0,& 5\mid n,\\ 1,& n\equiv 1,4(\mathrm{mod}5),\\ -1,& n\equiv 2,3(\mathrm{mod}5).\end{array}$$

It therefore contains the rational zeta response and an additional Dirichlet response. A use of the source’s GRH premise for this field already assumes RH for the rational zeta factor. The recurrence and the two real embeddings do not supply that premise.

The existing golden prime classification organizes rational primes by these split, inert and ramified classes. The local prime-ideal norms are $p,p$ for a split prime, $p^2$ for an inert prime, and $5$ for the ramified prime. In particular a norm cutoff $x$ includes inert rational primes only through $\sqrt{x}$.

To preserve this cutoff in a comparison, put

$${\mathcal{P}}_{\mathbb{Q}}(x)=\prod _{p\le x}(1-p^{-1}{)}^{-1},{\mathcal{L}}_5(x)=\prod _{p\le x}(1-{\chi }_5(p)/p{)}^{-1}.$$

Reindexing the same finite Euler factors gives, for $x\ge 2$,

$${\mathcal{P}}_K(x)={\mathcal{P}}_{\mathbb{Q}}(x){\mathcal{L}}_5(x)\prod _{\begin{array}{c}\sqrt{x}<p\le x\\ {\chi }_5(p)=-1\end{array}}(1-p^{-2}).$$

This is finite bookkeeping using the classical factorization and splitting data, not a new Mertens theorem. The inert correction and the signed ${\mathcal{L}}_5$ term must both be retained. Neither the field product nor its bias is an upper bound for the original signed Robin integral by itself. The current same-source requirement still concerns $A=\log N$ and $I_{\psi }(A)>-D^{\ast }(A)$ at the actual selected integer. An ideal-norm threshold or a logarithmic density does not identify that integer or pay this inequality.

The ideal observation loses the golden recurrence

In the existing FIB composition model, $M$ is multiplication by the unit $\theta$, $J$ is golden conjugation and $C=MJ$. For every integral ideal $I$ the induced images are therefore

$$M(I)=\theta I=I,J(I)=\overline{I},C(I)=\theta \overline{I}=\overline{I}.$$

Thus the four-step element rotation has only the conjugation action after passage to ideals; its square acts as the identity there. The same ideal represents all principal elements ${\theta }^{j}z$. The source’s ideal norm cannot recover the FIB window depth or the quantity observer $q(a,b)=2a+3b$ from that ideal alone.

The existing translation by $\alpha =1$ gives a concrete fiber test. The elements $1$ and $\theta$ generate the same unit ideal, but

$$(1+1)=(2)\ne {\mathcal{O}}_K,(\theta +1)=({\theta }^2)={\mathcal{O}}_K.$$

Consequently this element translation does not descend to an operation on principal ideals. These are applications of the existing unit and ideal relations, not a new no-go theorem or an obstruction to retaining the elements and their observers. A representation for the full affine FIB behavior must retain more than the ideal response.

The field supplies a genuine classical prime splitting slice and an independent $L(s,{\chi }_5)$ response. Its known conditional estimates, unit quotient and finite-cutoff accounting specify what a bridge back to Robin would have to retain; they provide no new signed estimate, critical-source coverage or proof of RH. All applications in this note are at paper level, without new Lean declarations or an originality claim.