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I Prime number inequality

  1. Feb 6, 2017 #1
    I believe this is probably a high level undergraduate question, but i could easily be underestimating it and it's actually quite a bit higher than that.

    I'm reading the Prime number theorem wikipedia page and i'm in part 4 under Proof sketch where sometime down they give in inequality:

    x is a natural number, p is for prime's obviously: ##\sum_{x^{1-\epsilon }\geq p\geq x}^{} \log p \geq \sum_{p\leq x}^{} logx##

    Where epsilon is any value greater than O. (O is some special value that they use in computer science a lot apparently, I might need to understand this value better to understand this inequality, i'm not sure.

    Can somebody help me understand this inequality? C'mon I know there are some really smart people here!
     
  2. jcsd
  3. Feb 7, 2017 #2
    Hi PsychonautQQ:

    I am not into number theory, but I am almost certain that the "O" is zero, i.e., "0".

    Regards,
    Buzz
     
  4. Feb 7, 2017 #3

    fresh_42

    Staff: Mentor

    My guess is ##O(something)##, e.g. ##O(1)\, , \,O(\log x) \, , \, O(\log \log x)## or even ##o(1)##. In any case I assume there is more than one typo in it, e.g. the ##\log x## in a summation over ##p## looks suspicious.
     
  5. Feb 7, 2017 #4

    mfb

    User Avatar
    2016 Award

    Staff: Mentor

    Here is a link to the article, the inequality discussed is the 6th formula in this section.
    "for any ε > 0" is a comparison with the number 0. The big O notation (with the letter O) is mentioned because it is used later in the same line.
     
  6. Feb 7, 2017 #5
    I found the equation in post #1 in the the linked article, although the post #1 version seems to have omitted a detail. It seems clear that the intent is
    ε > zero​
    and
    O(x1-ε)​
    is using the "big O" notation.

    Regards,
    Buzz
     
    Last edited: Feb 7, 2017
  7. Feb 7, 2017 #6

    mfb

    User Avatar
    2016 Award

    Staff: Mentor

    I thought it is the 6th formula there, but the right hand side is not identical.
     
  8. Feb 7, 2017 #7
    Wow, I am so smart for confusing 0 and O haha.

    I still don't understand the inequality though, can somebody explain why it makes sense intuitively or link a proof to me?
     
  9. Feb 8, 2017 #8

    mfb

    User Avatar
    2016 Award

    Staff: Mentor

    With the correct inequality from the Wikipedia article: We know ##x^{1-\epsilon} < p##. Take the logarithm on both sides and you get the inequality used in the article.
     
  10. Feb 8, 2017 #9
    And we know that x^(1-epsilon) < p because of the domain which is being summed over? p is required to be bigger?
     
  11. Feb 8, 2017 #10

    mfb

    User Avatar
    2016 Award

    Staff: Mentor

    We know it because the sum only runs over p satisfying this condition, right.
     
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