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Questions in number theory

  1. Aug 5, 2004 #1
    let note f(x)=O(g(x)) this f(x)<MG(x) being M a constant then would it be true?..

    If f(n)=o(n^u) then Sum(1<n<x)f(n)=O(n^u+1) adn Int(1,x)dnf(n)=O(n^u+1)

    Another question let be a(n)n^-s and b(n)n^-s two Dirichlet series so a(n)<b(n) for each n then if b(n)n^-s converges for a number Re(a)>1/2lso the series a(n)n^-s converges for Re(a)>1/2
     
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  3. Aug 7, 2004 #2

    shmoe

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    The n in your bounds for the sum and integral should be an x. Also, when u=-1, you get log(x) for a bound, not a constant.

    Hi, assuming both sequences are non-negative, then this looks fine.
     
  4. Aug 9, 2004 #3
    Another question let be F(x)=Sum(n<x)1/n^rthen does exist an r so:

    F(x)=O(x^1/2-r)?..where i could find a proof of that?...thanks.
     
  5. Aug 9, 2004 #4

    shmoe

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    No there doesn't. If r is not 1 then:

    [tex]\sum\limits_{n<x}\frac{1}{n^r}=O(x^{1-r})[/tex]

    So your asking if there is an r where [tex]1-r\leq 1/2-r[/tex]
    Which is of course false.

    If r=1, then again, no luck since log(x) is not [tex]O(x^{-1/2})[/tex].
     
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