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Robin’s criterion on divisibility

2022; Springer Science+Business Media; Volume: 59; Issue: 3 Linguagem: Inglês

10.1007/s11139-022-00574-4

1572-9303

Frank Vega,

Advanced Mathematical Identities

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