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Math Help - PNT

  1. #1
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    PNT

    I have managed to do part (a) and used Abel's Identity to prove part (b), but I don't know how to proceed with part (c).

    Any help would be great here.
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  2. #2
    MHF Contributor chiph588@'s Avatar
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    Ok, from part b we're given  g(x) = f(x) \log(x) - \int_{2}^{x} \frac{f(t)}{t} dt .

    Now let  x tend to infinity on both sides.
    You need to show  \int_{2}^{x} \frac{f(t)}{t} dt = o(x) as  x \to \infty . Then we know  f(x) \sim \frac{1}{4}\pi(x) \sim \frac{1}{4} \frac{x}{\log(x)} and we're done.

    Notice since  f(x) \sim \frac{1}{4}\pi(x) \;,\;\; \frac{f(x)}{x} \sim \frac{1}{4}\frac{\pi(x)}{x} \sim \frac{1}{4}\frac{1}{\log(x)} . So  \int_{2}^{x} \frac{f(t)}{t} dt \sim \frac{1}{4}\int_{2}^{x}\frac{1}{\log(t)} dt . This now becomes easier to verify as  o(x) .
    See here http://en.wikipedia.org/wiki/Logarit...otic_expansion.
    Last edited by chiph588@; July 21st 2009 at 02:31 PM.
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  3. #3
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    I've searched the net and I still can't seem to verify that the integral is little oh of x.

    Can anybody help here?
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