# Prime Number Theorem

• March 22nd 2009, 07:21 AM
funnyinga
Prime Number Theorem
Can anyone show me how to work this?

Consider the set of all prime numbers less that 80000. Estimate, using the Prime Number Theorem, the proportion which are less than 40000 and the proportion which are less than 8000.
• March 22nd 2009, 08:45 AM
chisigma
There are two simple approximations of the $\pi (n)$ function, which gives the number of primes less or equal to $n$

a) $\pi(n) \approx \frac{n}{\ln n}$ proposed by Gauss

b) $\pi(n) \approx \frac{n}{\ln n - 1,08366}$ proposed by Legendre

If we use a) is…

$\pi(80000) \approx 7086$ , $\pi(40000) \approx 3774$ , $\pi(8000) \approx 890$

If we use b) is…

$\pi(80000) \approx 7838$ , $\pi(40000) \approx 4204$ , $\pi(8000) \approx 1012$

Kind regards

$\chi$ $\sigma$