Results 1 to 2 of 2

Thread: Inequality with the prime-counting function

  1. #1
    Junior Member
    Joined
    Oct 2008
    Posts
    67
    Awards
    1

    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:

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

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

    Thanks in advance,
    HTale


    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Opalg's Avatar
    Joined
    Aug 2007
    From
    Leeds, UK
    Posts
    4,041
    Thanks
    10
    Quote Originally Posted by HTale View Post
    I'm going through a paper, and there is one statement which is:

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

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

    Take logs, and it says $\displaystyle \pi(n)<\frac n{\ln n}\ln3$. But ln(3)>1, so the result will follow from the prime number theorem.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Riemann's Prime Counting Function
    Posted in the Math Challenge Problems Forum
    Replies: 1
    Last Post: Jul 7th 2010, 10:51 AM
  2. Prime-counting function
    Posted in the Math Challenge Problems Forum
    Replies: 2
    Last Post: Oct 25th 2009, 12:10 AM
  3. Riemann's prime counting function
    Posted in the Number Theory Forum
    Replies: 1
    Last Post: Apr 8th 2009, 04:31 PM
  4. Counting number of solutions for an inequality
    Posted in the Discrete Math Forum
    Replies: 2
    Last Post: Mar 13th 2009, 04:28 PM
  5. Prime counting function
    Posted in the Number Theory Forum
    Replies: 4
    Last Post: Jun 11th 2006, 02:29 PM

Search Tags


/mathhelpforum @mathhelpforum