Zeta of "1"
- Thread starter BrianHarwarespecialist
- Start date
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
$$ \gamma \approx \frac{\pi}{\sqrt{2}} - \frac{\pi^2}{6}$$ 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
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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.
$$ \gamma \approx 1 - \sqrt{ \frac{328}{1835} } $$ 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.
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.