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March 20th, 2016, 09:24 PM   #1
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Generalizing the prime number theorem. Sorta.

I was thinking about the prime number theorem (because what else do I have to do with my time), and I ran across a problem I can't solve:

Say you have a function f(x), and the input is a natural number. The goal is to make f(x) act in such a way that it roughly estimates the number of factors that x has.

The reason I call this an extension of the prime number theorem is simply because you're trying to find information about the number of factors, rather than information about which numbers will have no non-trivial factors. So, I suspect that, in some way, the answer will turn out to be related to the proof of the prime number theorem.
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March 21st, 2016, 08:47 AM   #2
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Originally Posted by standardmalpractice View Post
I was thinking about the prime number theorem (because what else do I have to do with my time), and I ran across a problem I can't solve:

Say you have a function f(x), and the input is a natural number. The goal is to make f(x) act in such a way that it roughly estimates the number of factors that x has.

The reason I call this an extension of the prime number theorem is simply because you're trying to find information about the number of factors, rather than information about which numbers will have no non-trivial factors. So, I suspect that, in some way, the answer will turn out to be related to the proof of the prime number theorem.
Euler's totient function is pretty closed to what you are looking for.

-Dan
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