# Maximum error in the Prime Number Theorem

• Jul 12th 2010, 11:07 PM
blindConjecture
Maximum error in the Prime Number Theorem
I'm finding 2 conflicting answers for maximum value that pi(n)/(n/ln(n)) can output, where pi(n) is the prime counting function and ln(n) is the natural log.

An inequality by Chebyshev says that the maximum is less than 9/8 (can be found at Prime Number Theorem -- from Wolfram MathWorld). However, setting n=13 we get pi(13)/(13/ln(13))≈1.255 (type it in to WolframAlpha to see for yourself) which is greater than 9/8.

So my question is, is something about Chebyshev's statement wrong (I doubt it), or am I misunderstanding something, and if so, what?
• Jul 13th 2010, 07:50 AM
chiph588@
Quote:

Originally Posted by blindConjecture
I'm finding 2 conflicting answers for maximum value that pi(n)/(n/ln(n)) can output, where pi(n) is the prime counting function and ln(n) is the natural log.

An inequality by Chebyshev says that the maximum is less than 9/8 (can be found at Prime Number Theorem -- from Wolfram MathWorld). However, setting n=13 we get pi(13)/(13/ln(13))≈1.255 (type it in to WolframAlpha to see for yourself) which is greater than 9/8.

So my question is, is something about Chebyshev's statement wrong (I doubt it), or am I misunderstanding something, and if so, what?

I assume they meant to say for large enough $n$, like they do for the next inequality.
• Jul 13th 2010, 08:31 AM
blindConjecture
Yeah, after reading some more of that page I came to the conclusion. Thanks.