What percent of numbers are primes?

Question

I understand that there are infinitely many primes and (obviously) infinitely many integers, but is there any way to calculate the total percentage of integers that are primes? Thanks

If you ask it this way, the correct answer is $0%$, but you might be interested in a related question. Therefore, look at the prime number theorem — Stefan Mesken, Aug 13 '14 at 07:00

Darth Geek · Answer 1 · 2014-08-13T07:07:37.830

6

The function $\pi(n)$ tells us the amount of primes lower or equal to $n$. It is known that $\pi(n) \sim \frac{n}{\ln n}$ so the proportion as $n$ grows is about:

$$\frac{n}{\ln n}·\frac{1}{n} = \frac{1}{\ln n}$$

This means that as $n$ grows, the proportion tends to zero.

edited Aug 13 '14 at 07:07

answered Aug 13 '14 at 07:02

Darth Geek

12,543

Isn't it $\pi(n) \sim \frac{n}{\ln(n)}$? (Prime Number Theorem) – rehband Aug 13 '14 at 07:04
True, my bad. I'll correct it. – Darth Geek Aug 13 '14 at 07:06
Does this have anything to do with the circle constant π? – Someone Mar 16 '23 at 15:56

What percent of numbers are primes?

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