Loading ...
Sorry, an error occurred while loading the content.
 

Re: Numerology about the Apery Constant Zeta(3)

Expand Messages
  • djbroadhurst
    ... We should always try to do what most folk think can never be done. Had such an attempt succeeded, there would be several deeply unhappy Fields Medalists,
    Message 1 of 9 , Mar 14, 2011
      --- In primenumbers@yahoogroups.com,
      "WarrenS" <warren.wds@...> wrote:

      > I also attempted to use PSLQ to figure out whether Zeta(3)/Pi^3
      > was a low-degree low-height algebraic number.

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

      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
    • WarrenS
      ... --well, you (DJB) probably have better PSLQ and better hardware than I do (I bet) and could do so yourself... I see Broadhurst wrote a quantum field theory
      Message 2 of 9 , Mar 15, 2011
        > > Warren D Smith:
        > > 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.

        --well, you (DJB) probably have better PSLQ and better hardware
        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 "Grothendieck¬ĖTeichm: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.
      • djbroadhurst
        ... An editor solicited this article and I agreed on the strict condition that tedious be retained in the title. It happened that I needed to consult it
        Message 3 of 9 , Mar 15, 2011
          --- In primenumbers@yahoogroups.com,
          "WarrenS" <warren.wds@...> wrote:

          > Broadhurst wrote a quantum field theory paper
          > "where do the tedious products of zetas come from?"

          An editor solicited this article and I agreed on the
          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
        • WarrenS
          ... --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
          Message 4 of 9 , Mar 15, 2011
            > 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.


            --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.

            PSLQ has in fact discovered truths valid not to 12, but to
            an infinite number of decimals :)
          • mikeoakes2
            ... Even better: (exp(Pi*sqrt(163))-744)^(1/3) = 640319.999999999999999999999999390... See Cohen, CCANT, p.383, which explains where astonishing results such
            Message 5 of 9 , Mar 16, 2011
              --- In primenumbers@yahoogroups.com, "WarrenS" <warren.wds@...> wrote:
              >
              > --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.

              Even better:
              (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
            Your message has been successfully submitted and would be delivered to recipients shortly.