Primes vs. Natural Numbers

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xepher
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Primes vs. Natural Numbers

Postby xepher » Mon Dec 13, 2010 6:56 pm UTC

Does there exist a defined expression that describes the amount of prime numbers over the amount of natural numbers?

Is such an expression meaningless?
Also, if it helps, just assume the Riemann Hypothesis is true.

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Blatm
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Re: Primes vs. Natural Numbers

Postby Blatm » Mon Dec 13, 2010 7:00 pm UTC

Check out the Prime number theorem.

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Yakk
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Re: Primes vs. Natural Numbers

Postby Yakk » Mon Dec 13, 2010 7:01 pm UTC

There is a famous ratio of the number of prime numbers less than n to n. Is that what you are looking for?
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xepher
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Re: Primes vs. Natural Numbers

Postby xepher » Mon Dec 13, 2010 11:29 pm UTC

No, more like Cardinality of primes/Cardinality of Natural Numbers.

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Mo' Money
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Re: Primes vs. Natural Numbers

Postby ++$_ » Mon Dec 13, 2010 11:38 pm UTC

The cardinalities are obviously the same. You can't divide infinite cardinalities.

theorigamist
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Re: Primes vs. Natural Numbers

Postby theorigamist » Tue Dec 14, 2010 12:01 am UTC

It's pretty easy to see (using the Prime Number Theorem) that the Schnirelmann density of the primes is 0. Also, the natural density (also called asymptotic density) of the primes is 0.

skullturf
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Re: Primes vs. Natural Numbers

Postby skullturf » Tue Dec 14, 2010 12:06 am UTC

One possible way to try to make more precise the idea of the ratio (Amount of primes)/(Amount of natural numbers) is to look at what proportion of the first n natural numbers are prime, and then take the limit of that proportion as n approaches infinity.

http://en.wikipedia.org/wiki/Natural_density

For the primes, it turns out that you get 0. (And you don't need the full strength of the prime number theorem if you just want that fact.) But this answer by itself might seem a bit unsatisfying -- by itself, it doesn't tell you a whole lot about the primes and how they compare to other "thin" sets.

Edit: ninja'd!

xepher
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Re: Primes vs. Natural Numbers

Postby xepher » Tue Dec 14, 2010 12:25 am UTC

skullturf wrote:One possible way to try to make more precise the idea of the ratio (Amount of primes)/(Amount of natural numbers) is to look at what proportion of the first n natural numbers are prime, and then take the limit of that proportion as n approaches infinity.

http://en.wikipedia.org/wiki/Natural_density

For the primes, it turns out that you get 0. (And you don't need the full strength of the prime number theorem if you just want that fact.) But this answer by itself might seem a bit unsatisfying -- by itself, it doesn't tell you a whole lot about the primes and how they compare to other "thin" sets.

Edit: ninja'd!

DENSITIES. That's the thing I was looking for. Yeah, I expected it to be infinitesimal (aka 0).

theorigamist
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Re: Primes vs. Natural Numbers

Postby theorigamist » Tue Dec 14, 2010 5:58 am UTC

Yeah, I expected it to be infinitesimal (aka 0).

The density doesn't tell the whole story. There are still more primes than some other 0 density subsets. Here's something to consider. Denote the Schnirelmann density of a set A by d(A), and if A and B are two sequences, denote their sumset by A+B. A set A is called an essential component if for any set B that has 0<d(B)<1, we have d(A+B) > d(B). It turns out the primes are an essential component. That is, even though the primes have 0 density, you strictly increase the density of any positive density subset by adding the primes to it. Also, the primes form a basis for the natural numbers, which means if you add the primes to themselves some finite number of times, you can get any natural number. (This is very related to Golbach's conjecture.)

Just to convince yourself that these properties are special, if A is the set of powers of 2 (including 1), try to show that A is not a basis for the natural numbers, nor is it an essential component.

forgetful functor
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Re: Primes vs. Natural Numbers

Postby forgetful functor » Tue Dec 14, 2010 8:04 am UTC

For yet another way to think about "how many" primes there are among the natural numbers, recall that [imath]\sum 1/p[/imath] diverges. So in this sense there are many more primes than other density-zero subsets.


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