bibkey: kalyabin2026maximalgronwall authors: Gennadiy A. Kalyabin year: 2026 title: On Maximal Values of Gronwall Numbers for Integers with Given Greatest Prime Factor and Remainder in Modified Mertens Formula doi: null url: https://arxiv.org/abs/2603.24548v1 claim: The preprint states a sharp support-conditioned modified-Mertens calibration, with a lower construction on all sufficiently large supports and a matching upper bound only on its locally unimprovable support set E; it supplies no strict Robin sign or effective finite-source cutoff. strata_touched: [] license: citation-only triage: anchor

Support-conditioned maximal Gronwall numbers

The primary is arXiv:2603.24548v1, submitted 25 March 2026. The retrieved PDF has 16 pages and SHA-256 faead89924bd201d95a6ac827a9cc3e5784929d8b8800a13bd6797059ecabed9. The locators below use its printed pages. Theorem 1, Definitions 2–4, Theorem 2(I)(iv), and their arguments in §§2–4 were inspected against the versioned PDF and HTML. These are statements of a public preprint, not a journal-publication, complete proof-audit, or Lean-verification claim. No source text is vendored.

Exact support and quantifier scope

Put $P=p_k$, $T(P)={\prod }_{p\le P}p$, and

$$\begin{array}{rl}G(N)& =\frac{\sigma (N)}{N\log \log N},\\ {\tilde{W}}_k& =\{N>1:P^{+}(N)=p_k,T(p_k)\mid N\},\\ {\tilde{g}}_k& =\log \underset{N\in {\tilde{W}}_k}{\max }G(N),\\ S(P)& =\sum _{p\le P}\log \frac{p}{p-1},\\ Q(P)& =S(P)-\log \log \vartheta (P)-\gamma .\end{array}$$

The real logarithms in these comparisons are used on sufficiently large supports. Theorem 1, printed pp.2–3, states

$$\underset{k\to \infty }{\mathrm{liminf}}\sqrt{p_k}\log p_k[{\tilde{g}}_k-\gamma -Q(p_k)]=-2\sqrt{2}.$$

Its two parts have different domains. For each $\epsilon >0$ there is an existential cutoff $K_{\epsilon }$ such that part (I) gives

$${\tilde{g}}_k>\gamma +Q(p_k)-\frac{2\sqrt{2}+\epsilon }{\sqrt{p_k}\log p_k}(k>K_{\epsilon }).$$

This is obtained from a constructed $M_k\in {\tilde{W}}_k$; it is a bound on the maximum, not on every member of that support class. Part (II) gives

$$\log G(N)\le {\tilde{g}}_k<\gamma +Q(p_k)-\frac{2\sqrt{2}-\epsilon }{\sqrt{p_k}\log p_k}(k\in E,k>K_{\epsilon },N\in {\tilde{W}}_k).$$

Definitions 3–4, printed pp.5–6, specify

$$\begin{array}{rl}{U}_{1,k}& =\{N:P^{+}(N)=p_k,N\in GA1,G(Np)\le G(N)\text{ for every prime }p\le p_k\},\\ E& =\{k:U_{1,k}\ne \varnothing \}.\end{array}$$

Definition 2’s $U_1$ imposes the insertion comparison for every prime, including primes beyond the support. The paper reports both $E$ and its complement as infinite, referring to its earlier one-step-unimprovability preprint arXiv:1810.12585. The upper estimate is not an eventual bound over all support indices.

Directly reusable infinite one-step sources

The cited antecedent is Kalyabin, One-Step G-Unimprovable Numbers, arXiv:1810.12585v1, submitted 30 October 2018. Its PDF has 11 pages and SHA-256 6d45415c30ad326be291ccf4716d711975ea85bbabfbe822d119061d3898a72c. The definition on printed p.2, Theorem 1 on p.6, and Theorem 3 with its argument on p.10 were inspected. This is primary-source scope checking, not a complete proof audit or Lean verification.

Theorem 3 states that for every $M>0$ there is an actual integer $V_r$ with $P^{+}(V_r)>M$ and $V_r\in U_1$. Its one-step conditions retain the same integer:

$$G(V_r/p)\le G(V_r)(p\mid V_r),G(V_{r}p)\le G(V_r)(p\text{ prime}).$$

The 2026 paper’s Proposition 5(IV), in §3, recalls an infinite subsequence of these $V_k$ lying in $U_1$, hence with support indices in $E$. The existence of an unbounded family with both single-prime comparisons is therefore an existing supplier; neither its construction nor its infinitude needs another proof here. It is stronger source information than separately selecting a GA1 integer and an insertion-stable integer.

The multiplier comparison in $U_1$ still concerns one prime at a time. GA2 requires every positive integer multiplier. The existing workload note already records an actual one-step source improved by the joint multiplier $3\cdot 37$; that comparison is reused rather than recomputed. Thus the cited infinitude does not certify GA2 or establish an unbounded sequence of selected Robin-critical global maximizers. It also provides no signed modified-Mertens estimate or effective cutoff for one fixed source.

The antecedent’s Proposition 1(i) prints the opposite comparison $G(N)\le \min (G(N/p),G(Np))$. The conditions quoted above are those of its definition and Theorem 1; the reversed line is not used as a verified source condition.

Uniform clock expansion on the eligible sources

Theorem 2(I)(iv), printed p.8, states the uniform clock expansion

$$\underset{N\in U_{1,k}}{\sup }\left|\frac{\log N-\vartheta (p_k)}{\sqrt{\vartheta (p_k)}}-\sqrt{2}\right|⟶0(k\to \infty ,k\in E).$$

Its uniformity is over actual $U_{1,k}$ members. It does not identify $p_k$, $\vartheta (p_k)$, and $\log N$, or supply an effective cutoff for an independently specified finite integer.

Eligibility of the conditional critical source

Use the Caveney–Nicolas–Sondow source reduction directly. Under its counterexample hypothesis, let $N>5040$ be the least integer attaining the global maximum of $G$ over that range. This is the selected extraordinary, CA source, not the least Robin counterexample. GA1 and GA2 imply $N\in U_1$ and $N\in U_{1,k}$ with $k$ its own support index, so $k\in E$. Initial CA prime support gives $N\in {\tilde{W}}_k$.

The small-support conditions can be discharged for this selected source by reusing existing results. If $k\le 4$, all prime factors of $N$ lie in $\{2,3,5,7\}$. The repository’s seven-smooth Robin theorem, robin_seven_smooth, gives $G(N)<e^{\gamma }$, contradicting the selected counterexample-level maximum. Its exponents are unrestricted, so this exclusion has no finite enumeration cutoff. Hence $k>4$, and the endpoint conditions below give $\log N<p_{k+1}$ at this same integer.

Axler’s existing valuation stop, author version 2110.13478v3, Theorem 1.4 (published Theorem 3), gives strict Robin for $N>5040$ with $v_2(N)\le 20$. Thus $v_2(N)\ge 21$ at the selected source. If $k=5$, the endpoint would give $\log N<13$, whereas $\log N\ge 21\log 2>13$. Therefore $k\ge 6$, $P\ge 13$, and

$$T(P)\ge T(13)=30030>5040.$$

Every member of ${\tilde{W}}_k$ consequently belongs to the selected global maximization domain, and $\log G(N)={\tilde{g}}_k$ without an extra support-size hypothesis. This is a paper application of the cited source conditions and existing seven-smooth theorem, not a new valuation bound, enumeration, Lean declaration, or signed Robin estimate. The source’s asymptotic estimates still require their own $k>K_{\epsilon }$ cutoff; $k\ge 6$ does not pay that requirement.

Consequently the asymptotic calibration along such sources with unbounded support is

$$\log G(N)=\gamma +Q(P)-\frac{2\sqrt{2}+o(1)}{\sqrt{P}\log P},P=P^{+}(N).$$

This conditional asymptotic does not assert the existence of an unbounded critical-source sequence. A single selected global maximizer has fixed support; Theorem 1 gives no explicit $K_{\epsilon }$ certifying that it lies in the asymptotic range.

Reusing the signed-tail clock comparison

For an actual $N\in U_1$ with $P=P^{+}(N)=p_k$ and $k>4$, set $P^{+}$ equal to the next prime, and $A=\log N$. The 2026 Theorem 2(I)(ii)–(iii) and (III), together with the 2018 Lemma 3(i), already give

$$P<\xi (P,0)+\log P\le A\le \xi (P^{+},0)<P^{+}.$$

Here the largest-prime exponent is one by Theorem 2(I)(ii). Thus $\vartheta (A)=\vartheta (P)$; $(P,A]$ has no ordinary prime. This is direct use of the published one-step conditions, with no new source-selection theorem. The support restriction is material: the 2026 footnote 4 records $14\in U_1$, whereas $\log 14<7=P^{+}(14)$.

The signed-tail comparison in the Nicolas card, under “The endpoint condition at actual self-tangent sources”, can be reused with the current endpoints $P,A$. The uniform clock expansion above gives $A-\vartheta (P)=O(\sqrt{P})$ along these $U_1$ sources with support tending to infinity. The existing uniform short-interval input gives $A-P\le P^{+}-P\le P^{2/3}$ eventually. The classical prime-power decomposition and monotonicity therefore give the same interval budget used in that comparison, $|\psi (u)-u|=O(P^{2/3})$ for $P\le u\le A$. Direct integration against its existing kernel yields

$$I_{\psi }(P)=I_{\psi }(A)+O\left(\frac{P^{-2/3}}{\log P}\right),Z_{\psi }(P)=c_N{Z}_{\psi }(A)+O(P^{-1/6}),c_N=\frac{\sqrt{P}\log P}{\sqrt{A}\log A}=1+O(P^{-1/3}),$$

where $Z_{\psi }(t)=\sqrt{t}\log t{I}_{\psi }(t)$. The factor $c_N$ remains; ratio convergence alone does not bound the normalized tail. This is an application of the existing interval argument at the source’s own support, not the Nicolas card’s primorial cutoff or a selected sieve cutoff. It pays the eventual clock comparison on this source class, without supplying a signed lower bound, an effective cutoff for a fixed critical source, or an unbounded critical-source sequence. It introduces no new generic interpolation theorem, prime-gap estimate, or Lean claim, and does not extend the self-tangent pressure identity to arbitrary $U_1$ hosts.

What remains outside this supplier

The calibration leaves the needed one-sided comparison for $Q(P)$ open; it does not imply $\log G(N)<\gamma$. The eventual source-clock comparison above preserves its normalization factor and supplies neither that one-sided bound nor a finite-source cutoff. The support calibration, the classical exponent construction, and the existing pressure decomposition in the FIB volume are reused results, not new Robin estimates.

The paper identifies its lower construction as continuing arXiv:2201.02663, and its local source classes as continuing arXiv:1810.12585. These antecedents are citation links; their entire proofs are not independently certified here. The inspected proof text contains notation slips: §4.1’s verbal assignment of the two exponent thresholds reverses the order required by its products, and its claim that $\log \tilde{G}(t)$ tends to zero at infinity differs from the displayed formula. The formal theorem statements above are recorded with their stated scope; these slips are not silently repaired or used as verified intermediate lemmas.

The 2022 predecessors and the direction of their bounds

The earlier Remainder in Modified Mertens Formula and Ramanujan Inequality, submitted 7 January 2022, is the source already cited for the lower construction. Its main theorem, the §2 lemma (2.7), §3 and the final remarks were inspected in the versioned primary text. The lemma supplies a constructed integer at each sufficiently large support with a $2\sqrt{2}+o(1)$ loss; this is the predecessor of the 2026 lower calibration, not another construction to reproduce.

That paper’s main theorem already makes finiteness of ${\mathrm{limsup}}_{x\to \infty }Q(x)\sqrt{x}\log x$ equivalent to RH and records the consequence that the finite limsup is at most $2\sqrt{2}$. Its final remark announces narrower RH-dependent estimates for later papers. These are prior criteria and a conditional announcement, not an unconditional upper remainder bound.

The follow-on RH-Dependent Estimates of Remainder in Modified Mertens Formula, submitted 12 May 2022, states that narrower result. Its main theorem, §2’s Lemma 1, Remark 2 and equation (2.13), and §3’s Corollary 1 were inspected in the primary text and original TeX. The source archive has SHA-256 9ae03c336b0c7023231b7ea74b9a151ac57e43f64c43ab6ec0e3eeb7725a87e6. Both papers are preprints; their complete proofs are not independently certified, and no Lean verification is claimed.

For the same $Q(x)=S(x)-\log \log \vartheta (x)-\gamma$, $x\ge 3$, the May paper’s main theorem explicitly assumes RH and states

$$\underset{x\to \infty }{\mathrm{liminf}}Q(x)\sqrt{x}\log x\ge \frac{3}{2},\underset{x\to \infty }{\mathrm{limsup}}Q(x)\sqrt{x}\log x\le \frac{5}{2}.$$

These are limiting bounds, with no effective starting threshold for the selected critical integer supplied here. Remark 1 makes each bound separately equivalent to RH by the preceding Nicolas criteria. Neither the later support calibration nor a FIB coordinate change discharges that RH premise. The existing weighted-error supplier already retains the same distinction between conditional comparisons and an unconditional signed estimate.

The May paper also has an unconditional sign observation, but its direction matters. In its notation, define

$$H(x)=\underset{y\to \infty }{\lim }\left[\sum _{x<p\le y}\frac{1}{p}-\log \log \vartheta (y^{+})+\log \log \vartheta (x)\right],x>3.$$

For its primitive $\Phi (x)={\int }_0^x(\vartheta (t)-t)dt$, equation (2.9) defines $D(x)$ and $E(x)$ as boundary and integral terms linear in $\Phi$. Remark 2 states $H(x)\le D(x)+E(x)$ because its Taylor remainder is nonpositive. Equation (2.13) retains the exact correction

$$Q(x)+H(x)=-T(x),T(x)=\sum _{p>x}\sum _{m\ge 2}\frac{1}{m{p}^m}\ge 0.$$

Thus that observation supplies the lower comparison $Q(x)\ge -D(x)-E(x)-T(x)$, not the upper comparison needed for $Q(P)$ in the selected-source calibration above. The paper’s Corollary 1 obtains an RH criterion only after imposing $\Phi (x)=O_{\epsilon }(x^{3/2+\epsilon })$ for every $\epsilon >0$; the observation alone does not bound $D+E$. No such primitive bound or opposite-direction estimate is supplied here for the actual critical source.

The original TeX’s Proposition 4 prints $x<X_0$, whereas Proposition 5 uses $x>X_0$; equation (2.7) also prints $\Delta (x)$ inside an integral in $t$. These source inconsistencies are not silently repaired or treated as certified intermediate facts. The theorem statements, source-reported sign relation and their required direction are recorded for reuse, without a new Mertens estimate, Robin criterion or RH proof.