The empirical fit gives the power of ln(n) as slightly more negative than -1. For more information see the Wikipedia article on the prime counting function.
Sunday, May 27, 2007
Fraction of Primes 2
The fit for the fraction of numbers that are primes posted a week ago can be modified to approximate a known result by using ln(n) instead of log10(n) and rearranging terms somewhat to get,
The empirical fit gives the power of ln(n) as slightly more negative than -1. For more information see the Wikipedia article on the prime counting function.
The empirical fit gives the power of ln(n) as slightly more negative than -1. For more information see the Wikipedia article on the prime counting function.
Monday, May 21, 2007
The Fraction of Primes
A prime number is a number which is not devisible by any numbers except for one and itself.
I just did a curve fit for the fraction of prime numbers less than or equal to a given number and found a reasonably good fit shown in the plot below. This is probably not a new result since I once saw something similar for a bound on the number of primes. It is just the empirical fit that I found. The fraction of primes is the solid line and the fit the dotted line.
I just did a curve fit for the fraction of prime numbers less than or equal to a given number and found a reasonably good fit shown in the plot below. This is probably not a new result since I once saw something similar for a bound on the number of primes. It is just the empirical fit that I found. The fraction of primes is the solid line and the fit the dotted line.
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