Appendix No. 2: Alternative Proof of Robopol Theorem via Mertens

Abbreviations

• HCN: Highly Composite Numbers (maximize d(n)).
• SA: Superabundant numbers (Alaoglu–Erdős): σ(m)/m < σ(n)/n for all m < n.
• CA: Colossally abundant numbers (Erdős–Nicolas–Rankin): ∃ ε>0 with σ(n)/n^ε ≥ σ(m)/m^ε for all m≥1.

Theorem (Mertens):

The third Mertens theorem states that

which is equivalent to:

Defining beta(x) = Π (p/(p-1)) for p ≤ x, we get

However, this only tells us about the limit as \( x \to \infty \). For a strict inequality of the form beta(x) < e^γ log x above some threshold, we need an explicit version of the theorem that includes an error term.

According to explicit estimates in the literature (for instance, Rosser and Schoenfeld, 1962), there is a constant \( C \) and some \( x_0 \) such that for all \( x \ge x_0 \):

This is an explicit upper bound with a positive tail \(C/\log x\). By itself it does not imply \(\beta(x) < e^{\gamma}\,\log x\) for all large \(x\) without an additional compensating factor.

For an integer \( n=\prod p^{j} \), write

Introducing the deficit more explicitly, we may rewrite

In particular we always have the universal inequality \( \frac{\sigma(n)}{n}\le \beta(n)\,e^{-S(n)}. \)

A simple bound useful later is

since each term satisfies \(p^{-(j+1)}\le p^{-2}\) for \(j\ge1\).

We now pass to a stricter universal bound that keeps only the unit exponents.

Using the explicit Mertens bound at \(x=p_n\), we obtain

By \(\log(1-x)\le -x\) this is \(\le e^{\gamma}\,\log p_n\) once

Auxiliary bound B(n). Define \(B(n):=\beta(n)\prod_{p\in J_1(n)}\bigl(1-1/p^{2}\bigr).\) From the previous inequality we already have \(\sigma(n)/n\le B(n)<\beta(n)\). Whenever the j = 1 tail is non–empty we get \(\beta(n)-B(n)=\beta(n)\Bigl(1-\prod_{p\in J_1(n)}(1-1/p^{2})\Bigr).\) A more precise lower bound (sufficient for compensation) is \[\beta(n)-B(n)\;\ge\;\beta(n)\Bigl(1-\tfrac12\sum_{p\in J_1(n)}\tfrac1{p^{2}}\Bigr)\sum_{p\in J_1(n)}\tfrac1{p^{2}}.\] Since \(\sum_{p\in J_1(n)}1/p^{2}\approx C/(\log p_n)^2\ll1\), the bracket differs from 1 by less than a permille for relevant \(p_n\). Hence \(\beta(n)-B(n)\ge C/\log p_n\) once the tail condition is satisfied. In words: the unit-exponent tail already cancels the explicit Rosser–Schoenfeld surplus \(C/\log p_n\); the upcoming swap lemma is required only to guarantee that this tail is indeed present in every near-extremal profile.

Define the threshold \(T:=\log\!\bigl(1+ C/(\log p_n)^2\bigr)\). Pick the minimal discrete tail of primes above some \(y

Let \(r\) be the last prime with exponent \(\ge2\). For the factor contributions \(f(p,j)=\tfrac{p}{p-1}(1-p^{-(j+1)})\) define the increment \(\alpha_p(j)=\dfrac{f(p,j+1)}{f(p,j)}=\dfrac{1-p^{-(j+2)}}{1-p^{-(j+1)}}=1+\dfrac{(1-1/p)\,p^{-(j+1)}}{1-p^{-(j+1)}}\). Then \(\alpha_2(1)=7/6\) and \(\alpha_r(1)=1+1/(r^2-1)\le 9/8\) for all \(r\ge3\). Hence \(\alpha_2(1)/\alpha_r(1) \ge 28/27>1\).

If \(r>y\), perform a swap: decrease the exponent of \(r\) from 2 to 1 and increase the exponent of 2 by 1. The ratio \(\sigma(\tilde n)/\tilde n\) to \(\sigma(n)/n\) equals \(\alpha_2(1)/\alpha_r(1)>1\), and the j=1 tail gains \(r\), increasing \(\sum_{p\in J_1(n)}1/p^2\) by at least \(1/r^2\). Iterating while \(r>y\) strictly increases \(\sigma/n\), contradicting extremality. Therefore any extremal profile must satisfy \(r\le y\) and thus \(\sum_{p\in J_1(n)}1/p^2\ge T\).

Combining this with the explicit Mertens bound yields \( \frac{\sigma(n)}{n} \le e^{\gamma}\,\log p_n \).Using \( p_n < \log n \) (Appendix RH) we obtain \( \frac{\sigma(n)}{n} < e^{\gamma}\,\log\!\log n \).

Using the explicit Mertens bound, the strict upper bound via the j=1 tail, the discrete‑tail lower bound and the sharp swap lemma, we obtain

This yields Robin's inequality without invoking SA/CA assumptions; numerical checks in the companion scripts confirm the discrete‑tail step for large ranges.