Abstract
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...
Abstract
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}}).
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On Robin’s inequality
Article Open access 28 December 2022
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Acknowledgements
The author would like to thank Richard J. Lipton and Craig Helfgott for helpful comments and his mother, maternal brother and his friend Sonia for their support. The author also wishes to thank the referees for their constructive comments and suggestions.
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Vega, F. Robin’s criterion on divisibility. Ramanujan J 59, 745–755 (2022). https://doi.org/10.1007/s11139-022-00574-4
Received: 27 July 2021
Accepted: 08 March 2022
Published: 03 May 2022
Version of record: 03 May 2022
Issue date: November 2022
DOI: https://doi.org/10.1007/s11139-022-00574-4