Let be the number of positive divisors of the positive integer and let be Euler’s function. Then it is easily seen that

for each . So

holds for all .

Since there are infinitely many primes,

.

So it follows that

.

Therefore, by the sandwich theorem,

,

i.e., as . In layman’s terms, this is saying that the number of divisors of is small compared to when is large. The following plot of against for captures this nicely.

So this means, for example, that there are only finitely many positive integers with more than divisors, which I think is pretty neat!

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