Robin’s criterion on divisibility

Robin’s criterion states that the Riemann hypothesis is true if and only inequality $$\sigma (n) < e^{\gamma } \times n \log n$$ holds for all natural numbers $$n > 5040$$ , where (n)$$ sum-of-divisors function of $$\gamma \approx 0.57721$$ Euler–Mascheroni constant. We show Robin are not divisible by some prime between 2 1771559. prove when $$\frac{\pi ^{2}}{6} n' \le $$n>5040$$ $$n'$$ square free kernel number n. The possible smallest counterexample implies $$q_{m} e^{31.018189471}$$ $$1 \frac{(1 + \frac{1.2762}{\log q_{m}}) (2.82915040011)}{\log n}+ \frac{\log N_{m}}{\log n}$$ $$(\log n)^{\beta 1.03352795481\times (N_{m})$$ (2.82915040011)^{m} N_{m}$$ $$N_{m} = \prod _{i 1}^{m} q_{i}$$ primorial order m, $$q_{m}$$ largest divisor $$\beta \frac{q_{i}^{a_{i}+1}}{q_{i}^{a_{i}+1}-1}$$ an Hardy–Ramanujan integer form $$\prod _{i=1}^{m} q_{i}^{a_{i}}$$ .