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

Author: Ing. Robert Polák (Slovakia)

Date: 10.02.2025

This appendix provides an alternate route to proving the Robopol theorem using the third Mertens theorem and its explicit estimates (e.g., from Rosser–Schoenfeld), without employing smooth functions or \(\Delta x\)-based arguments.

1. Statement of the Robopol Theorem

In the main text, the Robopol theorem is formulated (in versions (3.1) and (3.2)) for highly composite numbers \( n \). Let \( p_n \) be the largest prime divisor of \( n \). Then, for sufficiently large \( n \):

\( \beta(n) = \prod_{p \le p_n} \frac{p}{p-1} < e^{\gamma}\,\log\bigl(p_n\bigr) \quad \text{(3.2)} \)

\( \beta(n) < e^{\gamma}\,\log\bigl(\log(n)\bigr) \quad \text{(3.1)} \)

depending on whether \(\log(n)\gt p_n\) or not. Our goal is to show these hold by applying an explicit form of the third Mertens theorem, without referencing a smooth function \(g(x)\).

2. Third Mertens Theorem (Asymptotic Form)

Theorem (Mertens):

The third Mertens theorem states that

\( \prod_{p \le x} \left(1 - \frac{1}{p}\right) \;\sim\; \frac{e^{-\gamma}}{\log x}, \quad \text{as } x \to \infty, \)

which is equivalent to:

\( \prod_{p \le x} \frac{p}{p-1} \;\sim\; e^{\gamma}\,\log x. \)

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

\( \beta(x) = \prod_{p \le x} \frac{p}{p-1} \sim e^{\gamma}\,\log x. \)

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.

3. Explicit Mertens Bound (Rosser–Schoenfeld)

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 \( x \ge x_0 \):

\( \prod_{p \le x}\frac{p}{p-1} < e^{\gamma}\,\log x + \frac{C}{\log x}. \)

For sufficiently large \( x \), the term \(\tfrac{C}{\log x}\) becomes arbitrarily small, ensuring a strict inequality:

\( \beta(x) < e^{\gamma}\,\log x \quad \text{for } x > x_0. \)

4. Transition to \( \beta(n) \) and Highly Composite \( n \)

Let \( n \) be a highly composite number with the largest prime factor \( p_n \). Then by definition:

\( \beta(n) := \prod_{p \le p_n} \frac{p}{p-1}. \)

Applying the explicit Mertens bound to \( x = p_n \) yields

\( \beta(n) < e^{\gamma}\,\log\bigl(p_n\bigr), \quad \text{as long as } p_n > x_0. \)

When \(\log(n) > p_n\) (common for highly composite numbers), it follows that

\( \beta(n) < e^{\gamma}\,\log(\log n). \quad \text{(3.1)} \)

Thus, both (3.1) and (3.2) of Robopol can be satisfied once \(n\) is large enough so that \( p_n \) exceeds the constant threshold \( x_0 \).

5. Precise Conditions on \(n\) and \(p_n\)

In the main text, Robopol theorem often states explicit values such as "If \( p_n \ge p_{10} \) (i.e., \( p_{10} = 29 \))" or "If \( p_n \ge p_{100} \)" etc., ensuring the function \( \beta(n) \) remains below \( e^{\gamma}\,\log\bigl(p_n\bigr) \). Once the explicit Mertens bound shows a threshold \( x_0 \approx 29 \) (or larger), the theorem holds for all greater \( p_n \).

Summarizing:

For large highly composite \( n \), if \( \log(n) > p_n \) and \( p_n \ge p_{10} \), then:
\( \beta(n) < e^{\gamma}\,\log\bigl(\log n\bigr). \)
If \( p_n \ge p_{100} \), the strict bound
\( \beta(n) < e^{\gamma}\,\log\bigl(p_n\bigr) \)
also holds (with even stronger numerical backing).

These correspond exactly to (3.1) and (3.2) in Robopol's stronger statements than Guy Robin's classical inequality.

6. Conclusion

By relying on the explicit Mertens theorem (e.g., Rosser–Schoenfeld), we obtain the strict inequality

\( \beta(x) < e^{\gamma}\,\log x \quad (x \gg 1) \quad\Longrightarrow\quad \beta(n) < e^{\gamma}\,\log(p_n) \quad(\text{or } e^\gamma\log(\log n)). \)

Substituting \( x = p_n \) and noting that \( n \) is highly composite with \( \log(n) > p_n \) (or other appropriate conditions), we immediately derive the Robopol theorem (3.1)/(3.2) without using smooth functions. Numerical verifications confirm these inequalities up to extremely large values of \( n \), consistent with the asymptotic and explicit forms of Mertens' theorem.

Tento dodatok prináša alternatívnu cestu k dôkazu Robopol teoremu na základe tretieho Mertensovho teorému a jeho explicitných odhadov (napr. Rosser–Schoenfeld), bez využívania hladkých funkcií alebo \(\Delta x\)-argumentov.

1. Formulácia Robopol teoremu

V hlavnom texte je Robopol teorem (verzie (3.1) a (3.2)) určený pre vysoko zložené čísla \( n \). Nech \( p_n \) je najväčšie prvočíslo, ktoré delí \( n \). Potom pre dostatočne veľké \( n \) platí:

\( \beta(n) = \prod_{p \le p_n} \frac{p}{p-1} < e^{\gamma}\,\log\bigl(p_n\bigr) \quad \text{(3.2)} \)

alebo

\( \beta(n) < e^{\gamma}\,\log\bigl(\log(n)\bigr) \quad \text{(3.1)} \)

v závislosti od toho, či \(\log(n)\gt p_n\) alebo nie. Cieľom je ukázať tieto nerovnosti priamo cez tretí Mertensov teorém, s explicitným chybovým členom.

2. Tretí Mertensov teorém (asymptotická forma)

Teorém (Mertens):

Tretí Mertensov teorém hovorí, že

\( \prod_{p \le x} \left(1 - \frac{1}{p}\right) \;\sim\; \frac{e^{-\gamma}}{\log x}, \quad \text{keď } x \to \infty, \)

čo je ekvivalentné:

\( \prod_{p \le x} \frac{p}{p-1} \;\sim\; e^{\gamma}\,\log x. \)

Definovaním beta(x) = ∏ (p/(p-1)) pre p ≤ x dostávame

\( \beta(x) = \prod_{p \le x} \frac{p}{p-1} \sim e^{\gamma}\,\log x. \)

Ide však o asymptotickú rovnosť (limit pri \( x\to\infty \)). Pre striktnu nerovnosť (napr. \(\beta(x) < e^{\gamma}\,\log x\)) nad istým prahom potrebujeme explicitnú verziu teorému s chybovým členom.

3. Explicitná Mertensova nerovnosť (Rosser–Schoenfeld)

Podľa explicitných odhadov v literatúre (napr. Rosser a Schoenfeld, 1962) existuje konštanta \( C \) a prah \( x_0 \) také, že pre všetky \( x \ge x_0 \):

\( \prod_{p \le x}\frac{p}{p-1} < e^{\gamma}\,\log x + \frac{C}{\log x}. \)

Pre dostatočne veľké \( x \) je \(\tfrac{C}{\log x}\) zanedbateľne malý, a tak dostaneme striktnu nerovnosť:

\( \beta(x) < e^{\gamma}\,\log x \quad (x > x_0). \)

4. Prechod na \( \beta(n) \) a vysoko-zložené čísla

Nech \( n \) je vysoko-zložené číslo s najväčším prvočíslom \( p_n \). Potom zo zvyklosti:

\( \beta(n) := \prod_{p \le p_n} \frac{p}{p-1}. \)

Uplatnením explicitnej Mertensovej nerovnosti na \( x = p_n \) dostávame

\( \beta(n) < e^{\gamma}\,\log\bigl(p_n\bigr), \quad \text{pokiaľ } p_n > x_0. \)

Ak ešte navyše platí \(\log(n) > p_n\) (typicky pri vysoko-zložených číslach), môžeme získať aj

\( \beta(n) < e^{\gamma}\,\log(\log n). \)

Tým priamo odvodíme (3.1) a (3.2) Robopol teoremu pre "veľké" \( n \).

5. Presné podmienky na \( n \) a \( p_n \)

V hlavnom texte Robopol teorem uvádza napr. "Ak \( p_n \ge p_{10} \) (tj. \( p_{10}=29\)) ..." alebo "Ak \( p_n \ge p_{100}\) ...". Týmito konkrétnymi prahmi sa zabezpečí, že:

beta(n) < e^γ log(p_n) (3.2) naozaj drží pre \( p_n \ge p_{10} \) či \( p_{100} \),
resp. beta(n) < e^γ log(log n) (3.1), ak tiež platí \(\log(n) > p_n\).

Explicitná Mertensova veta z Rosser–Schoenfeld garantuje, že nerovnosť \(\beta(x) < e^\gamma \log x\) "pre veľké \( x \)" platí, čo zodpovedá určitému prahu \( x_0 \). Potom stačí zabezpečiť \( p_n \gt x_0 \) a uvedená tvrdenia (3.1)/(3.2) sú splnené pre vysoko-zložené čísla.

6. Záver

Touto cestou (využitím explicitnej podoby tretieho Mertensovho teorému) sme ukázali, že

\( \beta(x) < e^{\gamma}\,\log x \quad (x \gg 1) \quad\Longrightarrow\quad \beta(n) < e^{\gamma}\,\log(p_n) \quad(\text{a/alebo } e^\gamma\log(\log n)). \)

Tým priamo vyplýva Robopol teorem (3.1)/(3.2) pre vysoko-zložené čísla, bez potreby hladkých funkcií a \(\Delta x\)-argumentov. Numerické testy (viď hlavný text) dokazujú, že takto odvodnená nerovnosť nebola prekročená ani pri extrémne veľkých hodnotách n, čo je v súlade s asymptotickým správaním Mertensovho súčinu.