bibkey: axler2024primorialcounting authors: Christian Axler year: 2024 title: Inequalities involving the primorial counting function doi: null url: https://arxiv.org/abs/2406.04018v1 claim: The paper applies Nicolas’s fixed-price finite low-benefit screening to arbitrary integers in a specific finite interval; that screening supplies neither a growing-price efficiency guarantee nor the FIB host’s signed Robin budget. strata_touched: [] license: citation-only triage: anchor

Inequalities involving the primorial counting function

Primary text: arXiv:2406.04018v1, version dated 6 June 2024. The relevant definitions and Lemma 5.2 are on printed p.7; Proposition 5.3 and its computation are on pp.7–8. Those statements have been inspected in this version. This note does not verify all proofs or rerun its Maple enumeration.

Fixed-price finiteness and the actual finite application

For a CA reference $C$ at parameter $\epsilon >0$, equation (5.5) defines the classical benefit

$${\mathrm{ben}}_{\epsilon }(n)=\log \frac{\sigma (C)}{C^{1+\epsilon }}-\log \frac{\sigma (n)}{n^{1+\epsilon }}=\epsilon \log (n/C)-\log \frac{Z(n)}{Z(C)}.$$

Lemma 5.2 states: for every fixed $\epsilon >0$ and every $\beta >0$, the set of positive integers $k$ satisfying ${\mathrm{ben}}_{\epsilon }(k)\le \beta$ is finite. The source explicitly attributes the result to Proposition 4.14 of Jean-Louis 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 original 2022 proof has not been inspected here: the publisher’s PDF endpoint returned an access page. The theorem statement recorded above is the exact restatement read in Axler’s primary text.

The following paragraph says the algorithm computing these sublevel sets is efficient when $\beta$ is not too large, specifically “not much larger than $\epsilon$”. This is the author’s qualitative efficiency statement, not a stated uniform complexity theorem.

Proposition 5.3 applies the method to all integers in the particular interval $(S{A}_{425},N_{39})$, where $N_{39}$ is the 39th primorial. With $\epsilon =0.00133$ and $\beta =0.000594$, the reported Maple intersection of the benefit sublevel set with this interval has the two elements

$$\frac{151}{149}S{A}_{425},\frac{157}{149}S{A}_{425}.$$

The proposition concludes that only the first has $Z(n)>8.8272$ in that interval. These are the source’s finite computations, not new certificates by this project. No superabundant hypothesis is imposed on the tested integer $n$; the reference $S{A}_{425}$ is certified CA by Lemma 5.1 at the chosen price.

Transported FIB threshold and the remaining host obligation

Use FIB §§230.2 and 234 with $y\to \infty$, $\ell =\log y$, $s=y\ell$, $\epsilon =1/s$ and $R=y/{\ell }^2$. For fixed $K>0$, the existing comparison gives, for every positive integer $d$ with $J_s(d)\le KR$,

$${\mathrm{Ben}}_{C_s,1/s}(d)\le \frac{J_s(d)+\log (C_s/\phi (C_s))}{s}\le {\beta }_y,{\beta }_y:=\frac{KR+\log (C_s/\phi (C_s))}{s}.$$

Here $C_s$ is the same-price CA reference. Thus the classical finite sublevel set at $(\epsilon ,\beta )=(1/s,{\beta }_y)$ contains the entire low-loss divisor set. The parameter ratio of this chosen uniform threshold, not the actual benefit of every $d$, is

$$\boxed{\frac{{\beta }_y}{\epsilon }=KR+\log (C_s/\phi (C_s))=\frac{Ky}{(\log y{)}^2}+O(\log \log y)⟶\infty .}$$

The source’s stated efficient-parameter regime consequently does not supply a uniform efficiency guarantee for this transported threshold. This does not prove that a particular enumeration is slow; its runtime has not been measured. Nor does finite cardinality at every fixed price give a bound uniform in $y$ or pay a signed Robin margin.

If $d\mid N$ is the low-loss witness for the actual host in §233.5, this upper bound concerns $d$. It does not state ${\mathrm{Ben}}_{C_s,1/s}(N)\le {\beta }_y$; the host cofactor requires its own transfer. The whole-host reconstruction application already carries that cofactor into the reduced host ratio, without enumerating the entire benefit sublevel set.

For the actual $N$, use the same-host loss certificate with $g=\gcd (N,C_s)$ and the actual resources $E_{+}=\log (N/g)$, $E_{-}=\log (C_s/g)$. Its lower bound still has to exceed

$$T_s(N)=\log {\mathcal{Q}}_s+\log N-s\log (e^{\gamma }\log \log N).$$

A finite whole-host screen can instead start from the actual signed budget, without transferring the divisor’s benefit. In the prescribed window $N\in [A,X]$, use $s=\log A\log \log A$ and $A>5040$. If that host fails the strict Robin test, then

$${\mathrm{Ben}}_{C_s,1/s}(N)\le \frac{T_s(N)}{s}.$$

The endpoint step in Axler’s proof directly supplies a uniform finite cutoff here. Writing $L=\log N\ge \log A$, the budget has derivative $1-s/(L\log L)\ge 0$ with respect to $L$, so $T_s(N)\le T_s(X)$. If $T_s(X)<0$, the nonnegative benefit already pays every host in this window. Otherwise any fixed positive cutoff

$${\beta }_{\mathrm{host}}\ge \frac{T_s(X)}{s}$$

contains every possible host failure. This includes the endpoint $T_s(X)=0$ by choosing a positive cutoff and retaining all zero-benefit ties. The parameter ratio for this host screen is ${\beta }_{\mathrm{host}}/\epsilon =s{\beta }_{\mathrm{host}}$; its behavior is not determined by the divisor cutoff ${\beta }_y$. After enumeration, the original window, Fibonacci residue, qualifying source divisor and exact strict budget still require their own checks. This is the source’s fixed-price endpoint method applied to the existing budget, not a new finiteness proof or a uniform tail certificate.

Lemma 5.2 and Proposition 5.3 supply neither that joint signed estimate nor an exclusion of all actual growing-price FIB hosts. Their definitions, finiteness theorem and reported numerical interval are reused directly; no new finiteness proof, Maple enumeration or RH conclusion is claimed.