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Math Help - Maximum error in the Prime Number Theorem

  1. #1
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    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?
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  2. #2
    MHF Contributor chiph588@'s Avatar
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    Quote Originally Posted by blindConjecture View Post
    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.
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  3. #3
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    Yeah, after reading some more of that page I came to the conclusion. Thanks.
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