bibkey: sabihi2026robinlagarias authors: Ahmad Sabihi year: 2026 title: Robin inequality, Lagarias criterion, and Riemann hypothesis doi: 10.33774/coe-2026-4mf39 url: https://arxiv.org/abs/1605.08273v14 claim: The v14 preprint claims an RH proof by combining Robin and Lagarias criteria with odd integer class-number sets; its key derivative-transfer proposition is false, so the claimed proof is not established. strata_touched: [] license: citation-only triage: anchor

Robin/Lagarias proof claim and its derivative-transfer gap

The source is arXiv:1605.08273v14, submitted 9 February 2026, and is also deposited as 10.33774/coe-2026-4mf39. The Cambridge record states that the manuscript is not peer reviewed. It partitions the integers into a finite computer-checked range and odd integer class-number sets, then claims Robin and Lagarias inequalities on every remaining class.

The key proposition is false as stated

Proposition 1 claims that, under positivity, differentiability, fixed sign and the absence of oscillation, $f=O(g)$ together with $g^{\prime }>0$ implies $f^{\prime }=O(g^{\prime })$. Monotonicity does not permit this derivative transfer. Let $\varphi$ be a nonnegative $C^1$ bump supported in $(-1/2,1/2)$ with $\varphi (0)=1$, and set

$$k(x)=\sum _{n\ge 1}{n}^2\varphi (n^4(x-n)),h(x)={\int }_0^{x}k(t)dt,$$

where the supports are disjoint. The integral of the $n$-th spike is $O(n^{-2})$, so $h$ is bounded and increasing, while $k=h^{\prime }$ is unbounded. With $g(x)=x$ and $f(x)=x+h(x)$, both functions are nonnegative, increasing, differentiable, and have no sign change or oscillation in the ordinary monotone sense; $f=O(g)$, but

$$f^{\prime }(x)=1+k(x)$$

is not $O(g^{\prime }(x))=O(1)$. Thus the proposition cannot justify the derivative estimate used later.

Consequence for the manuscript’s Robin route

The proof of Lemma 9 applies this proposition to a prime-counting asymptotic and writes an estimate for ${\pi }^{\prime }(x)$. The prime-counting function is a step function, and differentiating an $O$-term is not valid without an independent differentiable remainder estimate. Therefore the stated proof of the strict monotonicity of $R{O}_1$ is incomplete. Later large-number arguments cite Lemma 9 (and its consequences), so the manuscript does not establish its claimed all $n$ Robin inequality or RH proof.

This audit does not assert that the stated Robin or Lagarias criteria are false; it identifies a failure in this proposed proof. It also does not create a FIB/RH bridge: the manuscript works directly with ordinary prime products and the divisor sum, without transporting those quantities to the project’s $N_g=1+F_{r}g$ family.