Degree of Error in pi(x) Approximation

Date: 10/20/2004 at 10:36:31
From: Filippo
Subject: pi(x) approximation

How does the error in the formula x/ln(x) used to approximate pi(x)
(the primes counting function) behave by the growth of x?  I know that 
it gets smaller, but is there a formula that, using x, expresses the 
error on x/ln(x)?

I don't know anything else apart from Chebyshev limits, but they are 
the limits for ANY value of x, and I needed to know the upper limit as
x grows...

I tried for about a week to search the internet for an answer, but I 
had no success.

Date: 10/21/2004 at 11:15:30
From: Doctor Vogler
Subject: Re: pi(x) approximation

Hi Filippo,

Thanks for writing to Dr. Math.  You can get a lot of information 
about this from MathWorld at

  Prime Number Theorem
    http://mathworld.wolfram.com/PrimeNumberTheorem.html 

First of all, you said that the error at estimating pi(x) by x/ln(x)
gets smaller when x gets bigger, but this is only partially true.  The
*relative* error gets smaller, but the *absolute* error gets bigger. 
That means that

   pi(x)
  -------
  x/ln(x)

goes to 1 as x goes to infinity, and this is called the Prime Number
Theorem.  But of the difference

  pi(x) - x/ln(x)

this only says that it has order smaller than x/ln(x).  That is, it means

  pi(x) = x/ln(x) + o(x/ln(x)).

Are you familiar with big-O and little-o notation?

If the Riemann Hypothesis is to be believed, then we have their
equation (21),

  pi(x) = Li(x) + O(sqrt(x)*ln(x)),

where Li(x) is a smooth function (the logarithmic integral) which is
approximated by their equations (6) and (7),

  Li(x) = x/ln(x) + x/(ln x)^2 + O(x/(ln x)^3)

Since the amount by which Li(x) differs from x/ln(x) is approximately

  x/(ln x)^2,

and this is (asymptotically) much bigger than the error term between
pi(x) and Li(x), that means that

  pi(x) - x/ln(x)

is approximately

  x/(ln x)^2

for large x.  So Li(x) is really a much better approximation to pi(x),
and its error is probably as I stated above, though the proof depends
on the Riemann Hypothesis, which has not yet been proven.

If you have any questions about this or need more help, please write
back and show me what you have been able to do, and I will try to
offer further suggestions.

- Doctor Vogler, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 12/17/2004 at 07:10:44
From: Filippo
Subject: Thank you (pi(x) approximation)

Thanks for the time you spent to answer me, now that I studied
little-o and big-O notation I fully understand your answer; I hope to
go further on my personal conclusions, that obviously are o(what
mathematicians know)!

Thanks again.

- Filippo