Re: Old conjecture of Euler about Zeta(3) almost certainly wrong
- --- In firstname.lastname@example.org,
"djbroadhurst" <d.broadhurst@...> wrote:
> --- In email@example.com,Off list, Warren told me that both the typo
> "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."
(wrong power of log(2)) and also the error of judgment
as to what Euler concluded came from this popular account: