What is known of the function related to prime numbers pi(x),the number of primes under number x.?

Answer 1

It has been PROVED that as n tends to infinity, pi(x) tends to x / ln(x), or to Li(x) which is the "logarithmic integral", or to various other functions which all tend to each other. Hadamard, and Vallee Poussin, independently proved this in 1896. Erdos and Siegel proved it in 1950 without going into complex numbers.

The link below leads to some astonishingly precise functions for the actual values of pi(x).

Answer 2

The previous post had an error.
It is known that pi(x) is asymptotically equal
to x/log x. This is the famous prime number
theorem and was first proved independently
by Hadamard and De Valée Poussin in 1896
using the theory of analytic functions.
A more elementary proof was given in 1949 and 1950
by Erdös and Selberg.
For lots of information about the prime number theorem
consult
http://mathworld.wolfram.com/PrimeNumberTheorem.html .

Answer 3

I take position to attraction to close that this function is asymptotic to the logarithmic fundamental of x. That function is defined because the fundamental of one million/ln(x) taken as a lot as x, even with the undeniable fact that the lefthand certain varies from definition to definition. It in problem-free words introduces a small numerical distinction. besides, the actual incontrovertible reality that pi(x) ~ Li(x) replaced into shown by 2 human beings both around the initiating of the 1900s. the end result's called the accurate decision Theorem.

Answer 4

There are whole books written about this function. The most important thing about the prime function pi(x) is that it is asymptotically equal to x/log x, suggesting that the distribution of primes also has this form. This has not yet been proved, and is one of the greatest outstanding problems in mathematics.

So if you are looking for a good thesis problem, there's one for you!