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→Example: example with sigma_{-1} |
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where ''p'' denotes a prime.
In 1915, Ramanujan proved that under the assumption of the [[Riemann hypothesis]],
:<math>\ \sigma(n) < e^\gamma n \log \log n </math> (
holds for all sufficiently large ''n'' {{harv|Ramanujan|1997}}. The largest known value that violates the inequality is ''n''=[[5040 (number)|5040]]. In 1984, [[Guy Robin]] proved that the inequality is true for all ''n'' > 5040 [[if and only if]] the Riemann hypothesis is true {{harv|Robin|1984}}. This is '''Robin's theorem''' and the inequality became known after him. Robin furthermore showed that if the Riemann hypothesis is false then there are an infinite number of values of ''n'' that violate the inequality, and it is known that the smallest such ''n'' > 5040 must be [[superabundant number|superabundant]] {{harv|Akbary|Friggstad|2009}}. It has been shown that the inequality holds for large odd and square-free integers, and that the Riemann hypothesis is equivalent to the inequality just for ''n'' divisible by the fifth power of a prime {{Harv|Choie|Lichiardopol|Moree|Solé|2007}}.
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