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13938Re: [PrimeNumbers] the three limits of classical prime theory-> a contradiction

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  • Andy Swallow
    Nov 3, 2003
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      On Mon, Nov 03, 2003 at 09:56:54AM -0800, Roger Bagula wrote:
      > 1) Limit[Prime[n], n--> Infinity]=n*Log[n] ( Euler, I think)
      > 2) Limit[PrimePi[n],n->Infinity]=n/Log[n] (Hadamard)
      > 3) Limit[PrimePi[n]/CompositePi[n],n-> Infinity]=1/Log[n] ( from Ulam,
      > in both books I have by him)
      >
      > How 3) can be made to contradict 2):
      > PrimePi[n]+CompositePi[n]=n
      > PrimePi[n]/CompositePi[n]=PrimePi[n]/(n-PrimePi[n])=1/Log[n]
      > Solving for
      > PrimePi[n]=n/(1+Log[n])
      >
      > I'm puzzled by this which I found this morning.

      n/(log n) is asymptotic to n/(1+log n). Think about it.

      And 1) is certainly not due to Euler. 1) is an equivalent form of the
      prime number theorem, and so wasn't proved until Hadamard (and de la
      Vallee Poussin independently) proved it in 1896. 3) follows painfully
      obviously from 2), and so doesn't need to be quoted from anywhere.

      Before the PNT was proved, it was only known that pi(x) was of order
      x/log x, which is weaker than it being asymptotically x/log x.

      Andy
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