What is the Prime Counting Function?

[svigneshkumars]

What does this represent ??

pi(x)=x/log(x)

 

[tiny-tim]

pi(x)=x/log(x)

Hi svigneshkumars!

Can you show us the context that it comes from?

 

[uart]

pi(x)=x/log(x)

pi(x) is called the "prime counting function". It's value is the number of primes less than or equal to x. For some examples, pi(13)=6 because there are 6 prime numbers up to and including 13 (2,3,5,7,11,13); similarly pi(20)=8 and so on.

x/log(x) is just an assymptotic approximation to the prime counting function, so your expression shouldn't really contain an equals sign. Better would be to use approximately equal, or better still to state that the ratio of pi(x) to x/log(x) goes to 1 as x goes to infinity.

BTW. That log is a base e of course.

 

[robert Ihnot]

You can look this up on Wolfram, "The Prime Counting Function." There it is mentioned:

This relation was first postulated by Gauss in 1792 (when he was 15 years old), although not revealed until an 1849 letter to Johann Encke and not published until 1863 (Gauss 1863; Havil 2003, pp. 176-177).

Another historica nugget gone into is: The notation for the prime counting function is slightly unfortunate because it has nothing whatsoever to do with the constant . This notation was introduced by number theorist Edmund Landau in 1909 and has now become standard. In the words of Derbyshire (2004, p. 38), "I am sorry about this; it's not my fault. You'll just have to put up with it."