bibkey: axler2023robin authors: Christian Axler year: 2023 title: On Robin’s inequality doi: 10.1007/s11139-022-00683-0 url: https://arxiv.org/pdf/2110.13478v3 claim: The totient bound inside the proof gives a stronger joint prime-valuation stopping condition and an exact finite resolution for that condition. strata_touched: [] license: citation-only triage: anchor

On Robin’s inequality

The published article is Christian Axler, The Ramanujan Journal 61 (2023), 909–919. Formula numbers below refer to the inspected author version v3. Theorem 1.4 there corresponds to the published Theorem 3.

The bound needed for joint prime observations

Write $N_k={\prod }_{i=1}^k{p}_i$, $L=\log \log n$, and $a_0=0.0094243$. Lemma 2.3 states Robin for $5041\le n\le N_K$, where

$$K=999999476056,p_K=29996208012611.$$

Equations (3.4)–(3.5), in the proof of Theorem 1.3, imply

$$\frac{n}{\phi (n)}<e^{\gamma }\left(L+\frac{a_0}{L^2}\right)(n\ge N_K).$$

The proof’s analytic threshold is the smaller prime endpoint $29996161880813$, so the displayed finite and analytic ranges overlap. Retaining this bound on $n/\phi (n)$ is essential: multiplying a bound on $\sigma (n)/n$ itself by a local factor would not be justified.

The constant $ϵ=3.15367\cdot 10^{-7}$ used in Corollary 3.1 also satisfies $a_0/L^3<ϵ$ throughout $n\ge N_K$. An exact rational check of the overlap uses equation (3.3)’s cited theta estimate

$$\vartheta (x)>x\left(1-\frac{0.00033277}{(\log x{)}^2}\right).$$

At $x=p_K$, rational logarithm enclosures give

$$\log x>\frac{31032092090021}{10^{12}},\log \vartheta (x)>\frac{31032091744463}{10^{12}},$$

and $ϵ(31032091744463/10^{12}{)}^3-a_0>0$. The cutoff program certifies these arithmetic comparisons. It treats the published analytic bounds and finite verification as inputs; it does not rerun or formally verify them. The author version’s reference to Lemma 2.2 for the small range in this proof is not used here; the finite verification is Lemma 2.3.

For primes $P$ actually dividing the same integer $n>5040$, set

$${\eta }_P(n)=\prod _{p\in P}(1-p^{-(v_p(n)+1)}),T_A=\frac{10^{12}}{10^{12}+315367}.$$

Then ${\eta }_P(n)\le T_A$ implies strict Robin: above $N_K$, use

$$Z(n)\le {\eta }_P(n)\frac{n}{\phi (n)}<(1+ϵ){\eta }_P(n)e^{\gamma }L\le e^{\gamma }L;$$

below $N_K$, apply Lemma 2.3. An absent prime cannot contribute a factor. This is a direct application of the published proof, stronger than the Hertlein cutoff, with no new analytic theorem claim.

Complete five-direction classification and its resolution

Keep $P=(2,3,5,7,11)$ and lower exponents $b=(21,13,9,7,6)$, forced for a possible Robin counterexample by the previously cited individual stops. The exact report uses caps

$$c=(31,21,13,11,9).$$

The cap in each coordinate includes every greater exponent. All 9,900 cells are resolved: 1,144 have product supremum at most $T_A$ and 8,756 have product infimum greater than $T_A$. There are 42 minimal profiles outside the condition and 45 maximal certified regions. These counts are for this partition, not a density among natural numbers.

With $M=2^{21}{3}^{13}{5}^9{7}^{7}11^6$, the 42 minimal profiles correspond to the following divisibility antichain:

84, 210, 360, 378, 540, 882, 900, 1120, 1200, 1320, 1386,
1960, 1980, 2800, 3080, 3168, 3234, 3300, 3465, 4752, 7128,
7700, 10780, 14850, 15360, 16940, 17424, 26136, 33075, 35000,
59535, 138915, 294030, 385875, 1414875, 2223375, 3112725,
3189375, 5312384, 5740875, 66784256, 204526784

Thus a hypothetical counterexample must have $Mr\mid n$ for at least one listed $r$. This union is strictly contained in the 12-region union from the weaker cutoff, despite having more generators. For example, the exact profile of $6M$ is outside the Hertlein condition and inside the Axler condition. Membership in either remaining union does not imply a Robin violation. For authenticated $n=5040F_j$, the existing valuation transport gives $Dr\mid j$ for some listed $r$, where $D=2^{15}{3}^{10}{5}^8{7}^{5}11^5$; this alone does not incorporate other prime stops.

The caps also describe exactly the resolution needed by this predicate. Let $u_p=\min (v_p(n),c_p)$. The classifier is constant on every such cell, so these five clipped valuations determine whether ${\eta }_P(n)\le T_A$ on the full orthant. Equivalently, residues modulo $p^{c_p}$ supply sufficient data. This does not assert that the full residues are minimal states or that they determine Robin’s truth value in a rejected cell.

These caps are coordinatewise minimal among representations by clipped valuations: the maximum of coordinate $p$ over the 42 minimal outside profiles equals $c_p$. Pick a profile attaining it. That profile is outside, but decreasing its $p$ coordinate by one gives a certified profile by minimality. Any smaller cap in that coordinate identifies this pair, even if all other valuations are known exactly. Both profiles are realized by ordinary integers with those prime powers. This argument establishes the resolution of the supplied sufficient condition, not of primality or RH.

The FIB window recurrence can accumulate these modular observations from the same finite source. The contraction interval alone supplies no such valuation certificate. The useful bridge is the authenticated common integer plus its modular observations; neither the four-phase symmetry nor interval length is used as an Euler-factor weight.

A coupled new-prime slice on Fibonacci sources

The same bound also couples directions beyond the first five:

$$T_A-(1-13^{-6})(1-23^{-5})=\frac{1465648493004484688}{31067008116993059021656729}>0.$$

Hence $1\le v_{13}(n)\le 5$ together with $1\le v_{23}(n)\le 4$ already suffices for Robin, independently of all other prime factors. At the endpoint pair $(5,4)$, neither individual condition from Theorem 1.4 succeeds; the additional exclusion uses their joint product.

For the authenticated source $n=5040F_{Dk}$, the first divisibility ranks of $13,17,19,23$ are respectively $7,9,18,24$. At those ranks the Fibonacci values are $13,34,2584,46368$, each with prime valuation one. All four ranks divide $D$, and none of these four primes divides $5040D$. The rank/lifting formula cited in FIB §95 therefore gives

$$v_p(n)=1+v_p(k)(p\in \{13,17,19,23\}).$$

The individual stops force $Q=13^{4}17^{4}19^{4}23^3\mid k$. The new joint condition further forces $13^5\mid k$ or $23^4\mid k$. Thus, writing $k=Qt$, a hypothetical counterexample in this source family must have $13\mid t$ or $23\mid t$. Combining this with the 42 old-prime profiles gives the necessary union

$$DQrs\mid j\text{for some listed }r\text{ and some }s\in \{13,23\}.$$

The 84 generators are pairwise incomparable for divisibility: the 42 old-prime multipliers form an antichain and have no factor 13 or 23. This union is only an outer bound on candidates; multiplying all nine actual local factors can certify additional members of it. The argument uses one source $j$ for every direction, not separately selected local extrema. It neither gives a counterexample nor proves every source safe.

In fact all 84 lowest corners of these regions pass the full nine-factor product test. Their minimum rational slack is

$$\frac{3969942702286084876494714681732864182853322051528}{20579432895399995051033122675947739295611809721186640625}>0.$$

It occurs at old-prime profile $(22,15,9,8,7)$ and new-prime profile $(6,5,5,4)$. Holding these nine valuations exact permits arbitrary other prime factors, since discarded local factors are at most one. This does not certify the whole upper regions, where the nine valuations can grow. The corner calculation exposes the loss from testing groups separately; it is not evidence of simultaneous near-failure of Robin.

This note adds no Lean declaration, no new verified global Robin range, and no claim that the remaining 42 regions exhaust all known stopping rules.