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

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• 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 ]. OK, today
Message 1 of 4 , Jun 28, 2011
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 ].

OK, today we have an algorithm called PSLQ that can investigate such questions automatically. So I evaluated those 3 reals to 1050
decimals then ran it. It found:

175 * Zeta(3) + 89 * ln(2)^2 - 37 * ln(2) * Pi^2 = 3.129 * 10^(-6)

and running with 950 digits of precision, it found another approximate
relation with coefficients over 300 digits long. So anyhow...
Sorry Euler, looks like you were wrong -- probably no
integer relation exists, but if one does it is enormous.
(Euler, incidentally, had all those sorts of numbers memorized to large #decimal places and could do heavy arithmetic mentally.)

So then, what the hell, I tried PSLQ on
[ Zeta(3), ln(2)^2, ln(2)*Pi^2, Pi^3, ln(2), Pi, Pi^2, ln(2)^3, ln(2)*Pi, ln(2)^2*Pi, 1 ];
at 2000 decimal places. Again, best PSLQ finds incvolves coeffs with
179 digits or more.

Sorry Euler, again.
• ... 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
Message 2 of 4 , Jun 29, 2011
"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

By not checking the source, Warren made Euler look foolish
in a situation where in fact he had showed considerable wisdom,
by devising an ingenious method that makes an integer
relation improbable.

David
• Old conjecture of Euler about Zeta(3) almost certainly wrong WarrenS warren.wds@gmail.com warren_d_smith31 on Date: Tue Jun 28, 2011 7:42 pm ((PDT)) ...
Message 3 of 4 , Jun 29, 2011
Old conjecture of Euler about Zeta(3) almost certainly wrong
"WarrenS" warren.wds@... warren_d_smith31
on Date: Tue Jun 28, 2011 7:42 pm ((PDT))
said:

> 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)*Pi2 ].

What about [ ln(Zeta(3)), 2 * ln(ln(2)), ln(ln(2))+ 2 ln(Pi) ]?

Kermit
• ... 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 4 of 4 , Jul 2, 2011

> "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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