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Re: [PrimeNumbers] At last, a proof that RH is true

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  • Alan Eliasen
    ... I don t want to spend a lot of time on this, so I ll be brief and point out a couple major places where it s incorrect. The paper claims some amazing
    Message 1 of 3 , Jan 9, 2009
      Alex Petty wrote:
      > My colleague has settled the long outstanding question of Reimann's [sic]
      > Hypothesis and shown conclusively that all non trivial zeros of the
      > zeta function do indeed have Real part one half, ie. the hypothesis
      > has been proven to be true. To review this proof, now in pre-print,
      > please follow the url:
      >
      > http://www.singularics.com/science/mathematics/OnNeutronicFunctions.pdf

      I don't want to spend a lot of time on this, so I'll be brief and
      point out a couple major places where it's incorrect. The paper claims
      some amazing things, for example, "These functions are put together to
      reveal a new function whose difference from the Prime Number Function
      (2, 3, 5, 7, 11...) to infinity is zero..."

      This amazing function is very simple and can be summarized as the
      following:

      beta[x] := 6x + 1 (rearrangement of eq. 74)

      r[x] := 4x/5 - 15 (eq. 82)

      A[x] := 1 - 6x^x (eq. 84)

      u[x] := ln[-pi beta[x] A[x]] (eq. 83)

      J[x] := r[x] + u[x] (eq. 86)

      Where J[x] is supposed to be an approximation of the function that
      lists the primes, (I'll call it Prime[x]) e.g. 2,3,5,7,11. That is,
      Prime[1]=2 Prime[2]=3, etc.

      From this, I could reproduce the values in the table 27. No
      discrepancy there. The paper goes on to graph the first 140 terms of
      this to show how well J[x] matches Prime[x]. (I'll make a note, though,
      that when you're talking about the primes, looking only at the first 140
      terms of a sequence and extrapolating beyond that is a sure recipe for
      disaster.)

      But why only the first 140 terms? Probably because 64-bit IEEE-754
      floating-point hardware overflows after this! Note the x^x in eq. 84
      above, which grows rapidly. From these meager data points, the paper
      claims (eq. 88) that the maximum error J[x]-Prime[x] *for all x* is on
      the order of 12. This is completely wrong.

      If you use a real computing environment to evaluate larger numbers,
      the error does bounce around and stay smaller than 12 for very small
      values of x, but when you get to even x=206, the error is -19.6006. And
      then the error just starts to increase linearly (actually, a little
      worse than linearly.) At x=1400, the error is -398.109. At x=10000,
      the error is -4626.66. At x=20000, the error is -10667.6. And the
      trend continues. That's a lot larger than 12, but the paper says that
      "the maximum value of all error terms from 1 to infinity becomes very
      clearly" 12. That is no longer clear.

      Thus, the completely unsupported assertions about the error of this
      prime-approximating function in eq. 87-89 are wrong, as they seem to be
      based on incorrect ideas of how the functions actually behave. (And the
      leaps of faith from one unsupported equation to the next in this chain
      are immense.) There is of course no reason to believe the extrapolations
      to infinity, as the assertions are wrong for even very small numbers.

      I don't know if anything else in the paper relies on this, but it
      shows that the numbers weren't checked even to moderately-sized values,
      and assertions about the primes are incorrect.

      In addition, the definitions of things like the log integral (eq. 40)
      don't seem to be the definition of the log integral that I'm familiar
      with. It appears to be a summation of discrete terms, and not the
      integral! I may just not recognize it in this form, but it doesn't seem
      right at all. If that's important in the paper, it's another problem.

      My projection of the veracity of claims about the Riemann Hypothesis
      are thus approximately epsilon. But this is probably sufficient for
      this list.

      --
      Alan Eliasen | "Furious activity is no substitute
      eliasen@... | for understanding."
      http://futureboy.us/ | --H.H. Williams
    • Jeffrey N Cook
      Alan, Yesterday I accidently replied to only you. I was wondering if you could repost what I wrote to the group as well? Thanks, Jeff
      Message 2 of 3 , Jan 11, 2009
        Alan,

        Yesterday I accidently replied to only you. I was wondering if you
        could repost what I wrote to the group as well?

        Thanks,

        Jeff
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