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Re: Old conjecture of Euler about Zeta(3) almost certainly wrong

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  • djbroadhurst
    ... Off list, Warren told me that both the typo (wrong power of log(2)) and also the error of judgment as to what Euler concluded came from this popular
    Message 1 of 4 , Jul 2, 2011
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      --- In primenumbers@yahoogroups.com,
      "djbroadhurst" <d.broadhurst@...> wrote:

      > --- In primenumbers@yahoogroups.com,
      > "WarrenS" <warren.wds@> wrote:
      >
      > > Leonhard Euler [Opera Omnia Ser.1 vol.4 pp.143-144]
      > > conjectured that there was an integer linear relation amongst
      > > [ Zeta(3), ln(2)^2, ln(2)*Pi^2 ].
      >
      > No. The 3 terms were [Zeta(3), ln(2)^3, ln(2)*Pi^2]
      > and far from making a conjecture, Euler arrived at the
      > conclusion that no integer relation exists.
      >
      > A little scholarship revealed the following translation
      > of what Euler actually said, in the conclusion of
      > "De relatione inter ternas pluresve quantitates instituenda",
      > presented to the St. Petersburg Academy, 14 August 1775:
      >
      > "It would be superfluous to continue these operations further,
      > since one may now understand sufficiently well from here,
      > that no relationship between the three quantities taken is
      > given that is able to be resolved consistent with the truth.
      > Therefore, since I tried in vain to explore the investigation
      > of this reciprocal sum of cubes in different ways and this
      > method was called into use without result, it seems from
      > investigation that it rightly must be abandoned."
      >
      > http://eulerarchive.maa.org/docs/translations/E591trans.pdf

      Off list, Warren told me that both the typo
      (wrong power of log(2)) and also the error of judgment
      as to what Euler concluded came from this popular account:

      http://tinyurl.com/3eqzfdo

      David
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