bibkey: mishra2026finiterobin authors: Challenger Mishra; Rahul Sarkar year: 2026 title: A finite arithmetic form of Robin’s inequality and its equivalence to the Riemann hypothesis doi: null url: https://arxiv.org/abs/2609.26787v1 claim: The preprint introduces an omega(n)-truncated exponential Robin bound, proves it for several arithmetic classes, and claims equivalence with Robin’s inequality; it supplies no bridge from a FIB address to a bounded distinct-prime count. strata_touched: [] license: citation-only triage: anchor

A finite arithmetic form of Robin’s inequality

The source is arXiv:2609.26787v1, submitted 22 September 2026. This card records the stated theorem interfaces from that version. It is a preprint; the proofs and the Appendix B computations were not independently audited here, and no Lean verification is claimed.

The truncated criterion

For $n>5040$, put

$$P_K(x)=\sum _{j=0}^K\frac{x^j}{j!},C_{\omega }(n)=e^{\gamma }{P}_{\omega (n)}(\log \log \log n).$$

The paper calls

$$\frac{\sigma (n)}{n}<C_{\omega }(n)$$

the $\omega$-inequality. Since $P_K(x)<e^x$ for $x>0$, this is pointwise stronger than Robin’s bound. Theorem 1 claims the inequality unconditionally for $\omega (n)\le 6$ and for primorials; the paper derives the square-free and odd cases. Its Appendix B reports a finite check through $10^9$, which is source data rather than a project verification.

The same paper claims that the $\omega$-inequality is equivalent to Robin’s inequality, hence to RH, and that a minimal counterexample would have initial prime support, non-increasing exponents, and be superabundant. The CA restriction is a reduction of the claimed equivalent criterion, not a proof that any CA number is a counterexample.

Boundary for the FIB route

The truncation order is the distinct-prime count $\omega (n)$, while a FIB five-window address records additive Zeckendorf inclusion states. No statement in the source bounds $\omega (N_g)$ for the canonical FIB family $N_g=1+F_{r}g$, and the local labels $[null,2,3,25,5]$ do not encode the prime factorization of $N_g$. Thus the $\omega \le 6$ theorem cannot be applied to that family without a new arithmetic bound.

The source’s superabundant/CA reduction also requires the actual integer to be an extremizer for $\sigma (n)/n$. A FIB address, a congruence $N_g\equiv 1(\mathrm{mod}{F}_r)$, or a reversible composition coordinate does not establish that extremality. The result is therefore a candidate-class filter and an alternative Robin criterion, not the missing same-price bridge or a proof of RH.

Primary proof locators and their unpaid premises

In the versioned PDF, Lemma 4 supplies the $\omega (n)\le 6$ case; Theorem 1 concerns primorials. These are the precise separate locators for the two unconditional source claims described above.

Lemma 20, printed p.13, states that a hypothetical least counterexample to the $\omega$-inequality has $K=\omega (n)\ge 21$. It uses the author’s finite candidate check (L.5), with the candidate exponent bounds from Corollary 17. This is a statement about that least counterexample, not a stated factor-count bound on every arbitrary FIB integer $N_g$ or on a least Robin counterexample. Such an application must retain the source’s minimality condition for the same inequality.

For that same hypothetical least counterexample, Lemma 21, printed pp.13–14, sets $t=\log \log \log n$, ${\tau }_K=\log \log (6K\log K)$ and uses the preceding structural bound $\log n<6K\log K$ to obtain

$$\begin{array}{rl}{T}_K(t)& :=\log \log n-P_K(t)\\ & <E_K:=\frac{{\tau }_K^{K+1}}{(K+1)!}{\left(1-\frac{{\tau }_K}{K+2}\right)}^{-1}<\frac{6.4\times 10^{-15}}{\sqrt{\log n}}.\end{array}$$

This controls the truncation tail at the source’s specified integer. It supplies no unconditional lower bound for the Robin margin $\log \log n-e^{-\gamma }\sigma (n)/n$.

Theorem 4, printed p.16, proves the converse implication from RH by combining this tail bound with Lemma 23, equation (25): under RH, for every $n>5040$,

$$\frac{\sigma (n)}{n}<e^{\gamma }\left(\log \log n-\frac{0.095}{\sqrt{\log n}}\right).$$

The source attributes this input to Corollary 1.2 of Nicolas, The sum of divisors function and the Riemann hypothesis, The Ramanujan Journal 58 (2022), 1113–1157, DOI 10.1007/s11139-021-00491-y. The Nicolas original proof has not been inspected here; the existing source-access boundary also records that its publisher PDF endpoint returned an access page. This 2022 input is distinct from the unconditional envelope comparisons in Nicolas’s 2025 manuscript.

Thus the published equivalence proof uses an RH-conditional margin; the small truncation tail cannot be used to remove that premise. Theorem 4 uses Lemma 21, while the separate superabundance conclusion in Theorem 3 also uses Lemma 22 and its finite check (L.6). The author’s finite checks and supporting analytic estimates remain external premises; no notebook or old computation is rerun or certified by this note. This dependency map identifies the relevant source statements and quantifiers, with no new criterion, unconditional signed estimate, cofinal theta sign or Lean result.