bibkey: hertlein2018robin authors: Alexander Hertlein year: 2018 title: Robin’s inequality for new families of integers doi: null url: https://arxiv.org/pdf/1612.05186v2 claim: An explicit totient-ratio upper bound and a verified finite interval give a joint sufficient condition on several prime valuations. strata_touched: [] license: citation-only triage: anchor

Robin’s inequality for new families of integers

The published paper is Alexander Hertlein, Integers 18 (2018), A71. The author version v2, Lemmas 1–3, pp. 2–4, is the source inspected for the formulas below.

Put $Z(n)=\sigma (n)/n$ and $B(n)=e^{\gamma }\log \log n$. Lemma 2 states the unconditional strict bound

$$\frac{n}{\phi (n)}<\frac{1771561}{1771560}B(n)\text{for }n>c_0:=\exp (\exp (23.762143)).$$

Lemma 3 states Robin’s inequality for $5040<n\le 10^{10^{10}}$. The ranges overlap because $\log \log (10^{10^{10}})>23.8598>23.762143$. The first bound uses Hertlein’s explicit computation based on the algorithm of Akbary, Friggstad and Juricevic; the finite range uses Briggs and Robin’s interpolation between colossally abundant numbers. These published analytic and computational inputs have not been re-executed or formalized here.

Joint use of the local factors

Choose a finite set $P$ of primes that actually divide $n>5040$ and upper bounds $1\le v_p(n)\le a_p$. Define

$${\eta }_P(a)=\prod _{p\in P}(1-p^{-(a_p+1)}),T=\frac{1771560}{1771561}.$$

If ${\eta }_P(a)\le T$, the published results imply strict Robin for $n$. Indeed, for $n>c_0$, Lemma 1 and monotonicity in the exponents give

$$Z(n)=\frac{n}{\phi (n)}\prod _{p\mid n}(1-p^{-(v_p(n)+1)})\le {\eta }_P(a)\frac{n}{\phi (n)}<{\eta }_P(a)T^{-1}B(n)\le B(n).$$

For $5040<n\le c_0$, Lemma 3 applies. Equality in the rational condition is allowed because the totient bound is strict. An absent prime cannot be inserted as an extra factor in this argument. The paper’s proofs of Theorems 1–2 retain one local factor; retaining several factors is their direct joint application, with no claim of a new analytic theorem.

The exact five-direction core and its remaining boundary

For the exact core

$$M=2^{21}{3}^{13}{5}^9{7}^{7}11^6,{\eta }_M=\frac{72365886696479164830959537}{72365942756627122680000000},$$

the exact comparison is

$$1-T^{-1}{\eta }_M=\frac{15211459847969040463}{72365901907939012800000000}>0.$$

Consequently the literature already covers every $n=Mt$ whose remaining prime factors exceed 11, without a bound on their number or exponents. The finite support calculations in the FIB report are weaker certificates for this family; they do not extend the known Robin verification range. This statement requires the displayed core valuations to remain exact. For $C=M\cdot 2310$, all five valuations increase by one and

$$T^{-1}{\eta }_C-1=\frac{938607291436754075743}{2882283433033132800000000000}>0.$$

This failure is a limit of the sufficient condition, not a counterexample.

The accompanying exact program classifies the entire exponent orthant above $(21,13,9,7,6)$, in prime order $(2,3,5,7,11)$. Its caps $(25,15,11,9,7)$ have a precise meaning: the last bin in coordinate $p$ contains every exponent $a_p$ at least its cap $c_p$. On that bin the local factor lies in $[1-p^{-(c_p+1)},1]$. For smaller exponents the factor is exact. Multiplication gives a rational lower and upper bound for every bin, including its unbounded part.

All 270 bins are decided: 17 have product upper bound at most $T$; 253 have product lower bound greater than $T$; none straddles the threshold. Thus the finite computation classifies this entire unbounded exponent orthant for this particular sufficient condition. The 12 minimal vectors outside the condition are

(21,13, 9,9,7)  (21,13,10,8,6)  (21,14, 9,8,6)
(21,14,10,7,6)  (21,15, 9,7,7)  (22,13, 9,8,6)
(22,13,10,7,7)  (22,13,11,7,6)  (22,14, 9,7,6)
(23,13, 9,7,7)  (23,13,10,7,6)  (25,13, 9,7,6)

The product is increasing in each exponent. Every profile outside the condition dominates one of these vectors; decreasing any available coordinate of one of these vectors returns to the condition. In combination with the five separate valuation stopping rules, any hypothetical Robin counterexample must therefore be divisible by at least one of the 12 corresponding prime-power cores. This is a union of divisibility regions, not their intersection, and membership is only necessary.

Equivalently, with

$$\mathcal{A}=\{6,14,15,16,20,21,35,44,50,99,110,539\},$$

a hypothetical counterexample must satisfy $Mr\mid n$ for some $r\in \mathcal{A}$. These multipliers are obtained by subtracting the lower corner from each minimal exponent vector and taking the corresponding prime product. They form a divisibility antichain.

The lower corner uses Hertlein’s Theorem 2 for the primes 3, 7 and 11, and Christian Axler, On Robin’s inequality, The Ramanujan Journal 61 (2023), 909–919, Theorem 3, for 2 and 5. These are the five literature conditions already recorded in §86 of the FIB theory volume.

For an authenticated standard Fibonacci source $n=5040F_j$, $j\ge 3$, this also gives a smaller source-index search region without constructing $F_j$. The rank and lifting formulas cited in §95 of that volume give

$$D=2^{15}{3}^{10}{5}^8{7}^{5}11^5,j=Dk⟹(v_2(n),v_3(n),v_5(n),v_7(n),v_{11}(n))=(21,13,9,7,6)+(v_2(k),v_3(k),v_5(k),v_7(k),v_{11}(k)).$$

Thus any counterexample in this particular source family must satisfy $Dr\mid j$ for some $r\in \mathcal{A}$. If $D\nmid j$, a separate valuation stop already applies. This transport uses the actual source equality and the cited Fibonacci valuation formulas; a claimed index label alone is insufficient. Other published prime directions can still certify integers within these 12 index regions.

The result supplies a joint prime-resolution test for a common integer source. FIB window addresses may be used to compute its modular data, but the geometric interval lengths are not the weights in this Euler product. The unbounded tails already included in the bins do not settle the 12 remaining regions, where this particular local-factor criterion fails.

This note records an application of existing literature and an exact rational classification, not a new RH criterion or a new Lean proof. The retained result identifies the program bytes and the explicit parameters; outside never means that Robin’s inequality itself fails.