Zeta of "1"

Discussion in 'Physics & Math' started by BrianHarwarespecialist, Jun 11, 2015.

  1. Dr_Toad It's green! Valued Senior Member

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    Horsecrap.
     
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  3. rpenner Fully Wired Staff Member

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    There are an infinite set of numbers are close to any number. Nothing connects these approximations other than closeness to the chosen target, and human aesthetics.

    \( \gamma \approx \frac{\pi}{\sqrt{2}} - \frac{\pi^2}{6}\) Error: -7.08 × 10^-4 (Yours)

    \( \gamma \approx \frac{ \pi}{2 e}\) Error: 6.48 × 10^-4 (author unknown)

    \( \gamma \approx \frac{ \ln 9 }{\ln 45 }\) Error: -1.07 × 10^-5

    \( \gamma \approx \sqrt[4]{\frac{111}{10^3}} \) Error: -9.79 × 10^-6 (M. Hudson)

    \( \gamma \approx .1^{-\frac{.2}{.3-4^{-\left(.5^{-\left(\left(6 \times 7 \right)^{-.8}\right)} \right)}-.9}} = 10^{- \frac{1}{3+5 \times 2^{-2^{1+42^{-\frac{4}{5}}}}}} \) Error: -5.01 × 10^-10 (Gerrit de Blaauw's ordered pan-digital approximation)

    \( \gamma \approx \frac{2341}{ 1492 \, e}\) Error: -1.68 × 10^-10

    \( \gamma \approx \frac{ \ln \left( 9 + \frac{1}{4821} \right) }{ \ln \left( 45 - \frac{1}{729} \right) }\) Error: -2.03 × 10^-11

    \( \gamma \approx \frac{ 47712}{1069 (\pi^4 - e^3) }\) Error: -1.00 × 10^-11

    \( \gamma \approx .8^{.2674^9 - \frac{1}{3}} - .5 \) Error: -4.29 × 10^-12 (Richard Sabey's Pandigital approximation)

    \( \gamma \approx \ln \left( \frac{3671}{17458 }+ \frac{\pi}{2} \right) \) Error: -2.82 × 10^-12

    \( \gamma \approx \sqrt{\frac{1}{\pi} + \frac{3046}{204869} }\) Error: 9.93 × 10^-14

    \( \gamma \approx \frac{287443}{270955 \, \ln ( 2 \pi ) }\) Error: 6.59 × 10^-15

    \( \gamma \approx \frac{3731208726}{1587971713} - \sqrt{\pi}\) Error: -4.65 × 10^-22
     
    Last edited: Jun 24, 2015
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  5. rpenner Fully Wired Staff Member

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    \( \gamma \approx \frac{1}{2} \left( \zeta( 1 + \frac{1}{70000}) + \zeta(1 - \frac{1}{70000}) \right) \) Error: -9.89 × 10^-13

    \( \gamma \approx 1 - \sqrt{ \frac{328}{1835} } \) Error: -9.42 × 10^-13

    \( \gamma \approx \sqrt[5]{\frac{1332833}{136559}} - 1\) Error: -4.83 × 10^-15

    \( \gamma \approx \sqrt[3]{\frac{2394859}{66914}} - e\) Error: 1.43 × 10^-15

    \( \gamma \approx e - \sqrt[9]{ \frac{181762068}{192235}}\) Error: -2.95 × 10^-17

    \( \gamma \approx \pi - \sqrt[5]{ \frac{75428995}{680188} }\) Error: Error: 2.85 × 10^-17

    \( \gamma \approx \pi - \sqrt[6]{ \frac{2791879080}{9817601} }\) Error: Error: 1.17 × 10^-19

    \( \gamma \approx 1 - \sqrt[12]{\frac{25 \times 18934}{14512951059}}\) Error: 7.33 × 10^-22

    \( \gamma \approx \sqrt[12]{\frac{54427105845795}{7779823}} - \pi\) Error: -5.72 × 10^-24

    \( \gamma \approx \sqrt[12]{\frac{16397682819857}{9993800}} - e\) Error: 3.01 × 10^-24

    \( \gamma \approx \sqrt[11]{\frac{13654140881941}{7258095}} - \pi\) Error: 2.41 × 10^-24

    \( \gamma \approx \pi - \sqrt[9]{ \frac{3652901706851}{761730151} }\) Error: Error: 3.37 × 10^-25

    \( \gamma \approx \sqrt[5]{\frac{586319788321}{60072975364}} - 1\) Error: 4.74 × 10^-26

    \( \gamma \approx \pi - \sqrt[5]{ \frac{20551896232228}{185328641783} }\) Error: 7.16 × 10^-28

    \( \gamma \approx \sqrt[11]{\frac{193037381245234}{1284811801459}} - 1\) Error: -7.17 × 10^-31

    \( \gamma \approx \sqrt[10]{\frac{809275355576308143}{5356565981317}} - e\) Error: 1.29 × 10^-34

    \( \gamma \approx \frac{1}{2} \left( \zeta( 1 + 10^{-16}) + \zeta(1 - 10^{-16}) \right) \) Error: -4.85 × 10^-35

    \( \gamma = \lim_{x\to 0} \frac{1}{2} \left( \zeta( 1 + x) + \zeta(1 - x) \right)\)

    I hope this illustrates that the number of symbols in an approximation and the logarithm of the absolute error are roughly proportional when there is no connection between the form of the approximation and the targeted number. But when there is a connection, an exact formula can be written with a finite number of symbols.
     
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  7. BrianHarwarespecialist We shall Ionize!i Registered Senior Member

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    Yes that is correct but there is another reason I posted this, a reason I have not yet mentioned. That is becuase I have not yet taken the time to investigate it to my personal satisfaction.
     

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