Let and respectively be the number and the sum of the positive divisors of the positive integer . We will prove the following inequality:

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*First proof.* By the Cauchy-Schwarz inequality,

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*Second proof.* By Chebyshev’s inequality,

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*Third proof. *By the AM-GM inequality,

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*Fourth proof.* Again by the AM-GM inequality,

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*Fifth proof.* If factored into primes, then AM-GM gives

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We also trivially have . Hence for all ,

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In general, if is completely multiplicative, and

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then the first four proofs can be generalised to deduce that

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For example, if is the sum of the -th powers of the positive divisors of , then this shows that

for each . This, combined with the result from the previous post, gives

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i.e., as , for each .