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Re: Numerology about the Apery Constant Zeta(3)

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  • WarrenS
    I also attempted to use PSLQ to figure out whether Zeta(3)/Pi^3 was a low-degree low-height algebraic number. Result: If it has degree
    Message 1 of 9 , Mar 14, 2011
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      I also attempted to use PSLQ to figure out whether
      Zeta(3)/Pi^3
      was a low-degree low-height algebraic number.
      Result:
      If it has degree<=10 then its height is at least 10^91.

      So, sorry again:
      Zeta(3)/Pi^3 again fails to look unusual.
    • Jack Brennen
      No magic here, just the concept that you can find approximations to basically anything if you have enough degrees of freedom. For instance:
      Message 2 of 9 , Mar 14, 2011
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        No magic here, just the concept that you can find approximations to
        basically anything if you have enough degrees of freedom.

        For instance:

        (83/779)(605/689)^(3/2)Pi^3 == 2.71828182845...

        gives you the first 12 correct digits of e.

        (121/634)(305/724)^(3/2)Pi^3 == 1.6180339887...

        gives you the first 11 correct digits of the golden ratio (1+sqrt(5))/2.


        Note that there's an even better approximation to Zeta(3) at:

        (25/186)(86/197)^(3/2)Pi^3


        On 3/14/2011 9:59 AM, WarrenS wrote:
        >
        >
        > --- In primenumbers@yahoogroups.com, Sebastian Martin Ruiz<s_m_ruiz@...> wrote:
        >>
        >> I have obtained a curious
        >> aproximation to Apery constant Zeta(3)=
        >> 1.2020569031595942...
        >>
        >> (199/155)(16/165)^(3/2)Pi^3
        >
        > Hi.
        > To follow up on SMR's post and do some crude numerology, I computed Zeta(3)/Pi^3 to 1000 decimals:
        > 0.038768179602916798941119890318721149806234568039552579223126762123777137012286855271851...
        > and then computed its regular continued fraction expansion
        >
        > [0; 25, 1, 3, 1, 6, 3, 1, 1, 2, 1, 10, 3, 2, 1, 19, 3, 2, 1, 1, 3, 2, 3, 5, 3, 1, 1, 7, 1, 1, 1, 2, 1, 364, 11, 1, 84, 9, 34, 1, 7, 1, 63, 7, 1, 1, 4, 1, 5, 4, 7, 1, 1, 1, 5, 1, 4, 1, 5, 5, 9, 1, 21, 1, 9, 1, 1, 3, 2, 7, 1, 8, 5, 1, 7, 5, 2, 3, 1, 1, 1, 1, 1, 1, 1, 7, 8, 2, 2, 2, 1, 2, 1, 2, 6, 33, 99, 1, 1, 14, 1, 7, 2, 1, 1, 3, 4, 7, 1, 6, 5, 3, 1, 7, 8, 3, 5, 2, 1, 104, 1, 1, 1, 3, 8, 5, 1, 2, 1, 1, 11, 7, 1, 1, 1, 1, 2, 4, 1, 1, 1, 1, 1, 8, 1, 1, 7, 2, 4, 1, 1, 11, 1, 1, 1, 1, 35, 7, 1, 2, 3, 6, 1, 3, 1, 5, 81, 1, 2, 2, 7, 94, 2, 1, 2, 1, 3, 1, 2, 2, 1, 2, 1, 3, 1, 10, 75, 1, 2, 6, 3, 2, 7, 1, 1, 1, 6, 1, 4, 1, 1, 10, 1, 8, 2, 2, 1, 2, 1, 5, 6, 2, 1, 3, 1, 5, 1, 4, 3, 2, 5, 3, 3, 1, 10, 3, 1, 1, 9, 1, 1, 3, 3, 4, 1, 1, 2, 2, 1, 1, 15, 1, 15, 7, 4, 2, 1, 3, 1, 1, 1, 1, 1, 1, 1, 5, 8, 1, 1, 2, 1, 2, 4, 3, 1, 1, 3, 12, 2, 3, 1, 10, 2, 5, 19, 1, 4, 4, 3, 1, 3, 1, 6, 1, 7, 1, 4, 1, 1, 2, 7, 94, 7, 4, 2, 1, 4, 1, 1, 2, 1, 2, 11, 2, 2, 1, 24, 2, 2, 1, 1, 6, 1, 1, 5, 2, 2, 1,
        1, 1, 10, 1, 19, 1, 3, 1, 1, 1, 6, 1, 161, 1, 7, 1, 4, 5, 1, 5, 1, 2, 1, 4, 12, 1, 6, 2, 1, 7, 1, 4, 108, 1, 5, 1, 1, 3, 4, 4, 1, 1, 7, 1, 32, 1, 1, 1, 1, 2, 2, 3, 1, 1, 4, 2, 1, 4, 14, 1, 1, 2, 3, 16, 1, 3, 1, 2, 1, 1, 5, 1, 15, 2, 1, 2, 30, 1, 94, 7, 1, 1, 1, 1, 1, 1, 2, 2, 6, 6, 4, 2, 1, 4, 1, 5, 19, 1, 1, 1, 4, 8, 22, 1, 2, 3, 1, 5, 1, 1, 5, 2, 10, 6, 3, 4, 13, 1, 1, 1, 2, 1, 1, 97, 1, 1, 5, 2, 8, 1, 1, 2, 1, 3, 36, 1, 1, 2, 2, 1, 40, 1, 13, 3, 1, 1, 226, 1, 5, 3, 1, 7, 3, 19, 1, 20, 49, 1, 1, 33, 1, 7, 2, 1, 4, 7, 3, 7, 8, 1, 1, 55, 1, 17, 6, 1, 1, 1, 2, 1, 1, 5, 2, 8, 26, 3, 5, 6, 1, 2, 2, 2, 3, 1, 2, 2, 1, 33, 1, 3, 2, 2, 42, 1, 1, 1, 3, 2, 2, 1, 26, 1, 4, 7, 13, 7, 1, 29, 11, 1, 2, 1, 4, 1, 14, 3, 5, 2, 59, 6, 2, 2, 1, 103, 1, 8, 1, 3, 1, 2, 1, 3, 1, 3, 1, 1, 12, 3, 13, 2, 2, 1, 13, 1, 1, 40, 4, 1, 2, 3, 1, 3, 1, 1, 12, 1, 7, 1, 2, 1, 1, 49, 1, 2, 1, 3, 1, 3, 8, 1, 4, 1, 2, 5, 1, 3, 1, 7, 2, 2, 24, 3, 2, 3, 2, 11, 2, 1, 15, 1, 3, 54, 1, 1, 1, 1, 1, 8, 1, 5, 1, 2, 5, 6
        , 2, 7, 2, 1, 1, 8, 1, 1, 5, 1, 2, 2, 4, 3, 8, 2, 1, 1, 8, 43, 1, 1, 1
        > , 17, 1, 3, 4, 24, 1, 6, 2, 490, 6, 3, 1, 3, 8, 1, 1, 27, 2, 2, 1, 2, 1, 18, 5, 5, 1, 2, 2, 1, 2, 1, 6, 1, 1, 2, 1, 1, 1, 2, 2, 2, 1, 1, 1, 1, 1, 2, 2, 1, 3, 1, 2, 1, 2, 2, 2, 2, 2, 1, 14, 1, 2, 1, 1, 2, 3, 1, 1, 1, 1, 3, 4, 1, 1, 1, 2, 1, 1, 1, 6, 1, 4, 1, 3, 1, 4, 1, 6, 3, 1, 1, 1, 2, 2, 26, 1, 1, 1, 1, 1, 1, 29, 2, 2, 2, 21, 6, 1, 2, 4, 70, 1, 50, 2, 1, 2, 1, 1, 3, 1, 3, 4, 1, 3, 3, 4, 2, 48, 1, 2, 1, 16, 1, 2, 2, 1, 32, 329, 3, 4, 12, 1, 3, 10, 2, 2, 1, 1, 1, 1, 1, 3, 3, 7, 400, 1, 3, 7, 4, 2, 9, 2, 2, 41, 1, 2, 75, 1, 12, 4, 2, 47, 32, 3, 1, 1, 1, 1, 1, 1, 1, 2, 1, 3, 1, 4, 2, 1, 1, 2, 2, 2, 1, 23, 1, 112, 12, 1, 2, 1, 11, 5, 5, 3, 7, 1, 1, 1, 8, 1, 12, 1, 8, 1, 5, 4, 9, 1, 3, 1, 1, 3, 1, 5, 8, 1, 1, 4, 1, 2, 2, 1, 2, 38, 1, 2, 1, 2, 2, 38, 59, 1, 8, 4, 2, 1, 4, 2, 3, 1, 29, 19, 2, 369, 1, 1, 7, 1, 5, 9, 2, 3, 1, 1, 3, 2, 2, 34, 1, 1, 1, 11, 1, 3, 1, 8, 1, 10, 1, 1, 1, 32, 1, 3, 1, 1, 8, 4, 1, 1, 1, 2, 1, 3, 1, 1, 2, 1, 8, 2, 3, 1, 15, 1, 2, 2, 1, 10, 5, 1, 2, 2, 4, 2
        9, 8, 5, 1, 1, 2, 5, 1, 2, 1, 2, 6, 1, 1, 1, 11, 2, 5, 1, 2, 115, 6, 14, 4, 121, 1, 4, 2, 15, 34, 24, 6, 1, ...]
        >
      • 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 3 of 9 , Mar 14, 2011
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          --- 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 4 of 9 , Mar 15, 2011
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            > > 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 5 of 9 , Mar 15, 2011
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              --- 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 6 of 9 , Mar 15, 2011
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                > 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 7 of 9 , Mar 16, 2011
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                  --- 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
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