Relation To The Riemann Hypothesis
In the 1980s Guy Robin showed that the Riemann hypothesis is equivalent to the assertion that the following inequality is true for all n > 5040:
This inequality is known to fail at n = 5040, but Robin showed that if the Riemann hypothesis is true then this is the last integer for which it fails. The inequality is now known as Robin's inequality after his work. It is known that Robin's inequality, if it ever fails to hold, will fail for a colossally abundant number n, thus the Riemann hypothesis is in fact equivalent to Robin's inequality holding for every colossally abundant number n > 5040.
Read more about this topic: Colossally Abundant Number
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