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Math Help - Inequality with the prime-counting function

  1. #1
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    Inequality with the prime-counting function

    Hi there guys, I hope I've posted this in the right section;

    I'm going through a paper, and there is one statement which is:

     n^{\pi(n)} < 3^n

    for sufficiently large n, where \pi(n) is the prime-counting function defined as the function counting the number of prime numbers less than or equal to n. I'm having real difficulty justifying this claim, and I'd be really grateful if someone could help me out.

    Thanks in advance,
    HTale


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  2. #2
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    Quote Originally Posted by HTale View Post
    I'm going through a paper, and there is one statement which is:

     n^{\pi(n)} < 3^n

    for sufficiently large n, where \pi(n) is the prime-counting function defined as the function counting the number of prime numbers less than or equal to n.

    Take logs, and it says \pi(n)<\frac n{\ln n}\ln3. But ln(3)>1, so the result will follow from the prime number theorem.
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