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
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