- --- In primenumbers@yahoogroups.com,

"WarrenS" <warren.wds@...> wrote:

> I also attempted to use PSLQ to figure out whether Zeta(3)/Pi^3

We should always try to do what most folk think can never be done.

> was a low-degree low-height algebraic number.

Had such an attempt succeeded, there would be several deeply

unhappy Fields Medalists, who believe (as I do) in the

Drinfeld-Deligne conjecture that the Grottendieck-Teichmueller

algebra has precisely one generator in each odd degree.

Yet that is, emphatically, not a reason for avoiding serious

experiments to investigate a contrary hypothesis. To the best of

my recollection, the possibility that zeta(3)/Pi^3 might be an

algebraic number has been investigated to many tens of thousands

of decimal digits of numerical precision. However, I am unable

to provide a reference. Might someone else do so, please?

David > > Warren D Smith:

--well, you (DJB) probably have better PSLQ and better hardware

> > I also attempted to use PSLQ to figure out whether Zeta(3)/Pi^3

> > was a low-degree low-height algebraic number.

> DJ Broadhurst:

> Had such an attempt succeeded, there would be several deeply

> unhappy Fields Medalists, who believe (as I do) in the

> Drinfeld-Deligne conjecture that the Grottendieck-Teichmueller

> algebra has precisely one generator in each odd degree.

>

> Yet that is, emphatically, not a reason for avoiding serious

> experiments to investigate a contrary hypothesis. To the best of

> my recollection, the possibility that zeta(3)/Pi^3 might be an

> algebraic number has been investigated to many tens of thousands

> of decimal digits of numerical precision. However, I am unable

> to provide a reference.

than I do (I bet) and could do so yourself...

I see Broadhurst wrote a quantum field theory paper "where do the tedious products of zetas come from?" (actual title!) which mentions the "Drinfeld Deligne conjecture"

and "GrothendieckTeichm:uller algebra" (whatever they are) as well as using PSLQ.

Unfortunately non-experts (i.e. me) will have a difficult time understanding this paper.

Even after (trying to) read it, I still have almost no clue what DDC and GTA are,

and it gives zero cites to D,D,G and T's work.

Anyhow I agree Zeta(3)/Pi^3 being an "unusual number" was a pretty low-chance proposition, but SM Ruiz's post stimulated me to look, and sure enough, negative result.

About the idea that you can produce miracles on demand if you have enough fitting freedom... well, yes, but the approach I used attempts to quantify the miraculousness,

and the TRUE miracles yield far better approximations than the amount of fitting freedom would suggest. Don't accept lame miracles, demand real miracles.- --- In primenumbers@yahoogroups.com,

"WarrenS" <warren.wds@...> wrote:

> Broadhurst wrote a quantum field theory paper

An editor solicited this article and I agreed on the

> "where do the tedious products of zetas come from?"

strict condition that "tedious" be retained in the title.

It happened that I needed to consult it yesterday

and fortunately it was easy to google:> Ungefähr 84 Ergebnisse (0,18 Sekunden)

David

> About the idea that you can produce miracles on demand if you have enough fitting freedom... well, yes, but the approach I used attempts to quantify the miraculousness,

--for example (this is not due to me; perhaps it was first noticed by Ramanujan?)

> and the TRUE miracles yield far better approximations than the amount of fitting freedom would suggest. Don't accept lame miracles, demand real miracles.

exp(Pi * sqrt(n))

is within 10^(-12) of being an integer if n=163.

This is beyond what any kind of fitting contrivance would be expected to produce.

Also, no other n<25000 gets you even within a 100,000 times further away

from being an integer.

PSLQ has in fact discovered truths valid not to 12, but to

an infinite number of decimals :)- --- In primenumbers@yahoogroups.com, "WarrenS" <warren.wds@...> wrote:
>

Even better:

> --for example (this is not due to me; perhaps it was first noticed by Ramanujan?)

> exp(Pi * sqrt(n))

> is within 10^(-12) of being an integer if n=163.

>

> This is beyond what any kind of fitting contrivance would be expected to produce.

> Also, no other n<25000 gets you even within a 100,000 times further away

> from being an integer.

(exp(Pi*sqrt(163))-744)^(1/3) = 640319.999999999999999999999999390...

See Cohen, CCANT, p.383, which explains where astonishing results such as these come from.

Mike