bibkey: banks2007nicolasrobin authors: William D. Banks; Derrick N. Hart; Pieter Moree; C. Wesley Nevans year: 2007 title: The Nicolas and Robin inequalities with sums of two squares doi: null url: https://arxiv.org/abs/0710.2424v1 claim: A fixed positive-density restriction excluding exponent-one primes yields eventual Nicolas and Robin inequalities; the sum-of-two-squares Robin exceptions are bounded by 720. strata_touched: [] license: citation-only triage: anchor

The Nicolas and Robin inequalities with sums of two squares

The versioned author text, §1, equations (4)–(6) and Theorem 1, fixes a prime set $\mathcal{P}$ satisfying

$$0<\underset{x\to \infty }{\mathrm{liminf}}\frac{{\pi }_{\mathcal{P}}(x)}{\pi (x)}\le \underset{x\to \infty }{\mathrm{limsup}}\frac{{\pi }_{\mathcal{P}}(x)}{\pi (x)}<1.$$

Write $\mathcal{Q}$ for its complement and set

$$\mathcal{S}(\mathcal{P})=\{n\ge 1:p\in \mathcal{Q},p\mid n⟹p^2\mid n\}.$$

Theorem 1 states that all but finitely many members satisfy

$$\frac{n}{\phi (n)}<e^{\gamma }\log \log n.$$

Here $\phi (n)$ is Euler’s totient. The authors call this upper inequality the Nicolas inequality; its direction differs from the primorial Nicolas criterion recalled earlier in their introduction. Their §1 also records $\sigma (n)/n<n/\phi (n)$ for $n>1$, so these same members satisfy Robin. This consumes the published theorem, without reconstructing its density or minimization proof.

Corollary 2 gives a useful necessary profile: for every fixed reduced residue class $a(\mathrm{mod}m)$, all sufficiently large failures of their Nicolas upper inequality have a prime $p\equiv a(\mathrm{mod}m)$ with $p\mid n$ and $p^2\nmid n$. The same implication applies to Robin failures, by the preceding strict comparison. The exceptional threshold depends on the fixed class; the corollary supplies no uniform threshold when $m$ or the class varies with $n$.

The four-phase norm is an already solved scalar class

The Robin application immediately following Theorem 2 in §1 gives

$$n=a^2+b^2>720⟹\frac{\sigma (n)}{n}<e^{\gamma }\log \log n.$$

The article determines the complete exceptional values in this class. The classical two-squares characterization explains its prime-exponent restriction. Thus the four-phase invariant $E(a,b)=a^2+b^2$ of $C(a,b)=(-b,a)$ can use this bound directly; neither exception enumeration nor a new proof of that bound is needed.

The same theorem applies to golden norm magnitudes

For the golden integer $x=a+b\theta$, where ${\theta }^2=\theta +1$, write

$$Q(a,b)=a^2+ab-b^2,n=|Q(a,b)|>0.$$

Use the fixed set

$${\mathcal{P}}_5=\{5\}\cup \{p\text{ prime}:p\equiv 1,4(\mathrm{mod}5)\}.$$

The prime number theorem in these fixed progressions gives density $1/2$. Every complementary prime $p\equiv 2,3(\mathrm{mod}5)$ is inert in $\mathbb{Z}[\theta ]$. The classical inert-prime norm law gives $p\mid n⟹p^2\mid n$; this includes $p=2$. The repository’s golden carrier, conjugation, multiplicative norm and GoldenPrimeSplitting.golden_prime_of_mod_five_eq_two_or_three locate the algebraic inputs. Theorem 1 therefore applies to the same actual norm magnitude $n$ with $\mathcal{P}={\mathcal{P}}_5$.

Consequently all sufficiently large nonzero golden norm magnitudes satisfy the strict Nicolas upper inequality and Robin. The threshold is uniform over their representations because ${\mathcal{P}}_5$ is fixed; no numerical threshold or full golden-norm exception list is supplied here. This is a classical-source application, without a new Lean declaration, Lean verification or originality claim.

The norm readout does not replace the actual Robin integer

The FIB quantity is $q(a,b)=2a+3b$, with a separately retained unit bit when required. It is different from both $E(a,b)$ and $|Q(a,b)|$. To apply either norm-class bound to an actual Robin integer $N$, one must certify that $N$ itself belongs to that class; the norm of its FIB composition alone does not supply such a certificate.

The already covered FIB norm branches bound quantities $gq(a,b)$ under their stated source conditions. They are distinct from this bound on $|Q(a,b)|$, and this note does not extend those quantity families. Basic Fib recursion preserves $|Q|$, so increasing the recursion depth need not increase the norm magnitude or cross its threshold. Finitely many exceptional norm values can correspond to infinitely many compositions.

The actual CA right-tail source obligation still needs its directed signed estimate at the original source clock. No such estimate, covering certificate for that source family, or proof of RH follows from these norm-class bounds.