TOPICS
Search

Apéry's Constant Approximations


E. Pegg Jr. (pers. comm., Nov. 8, 2004) found an approximation to Apéry's constant zeta(3) given by

 zeta(3) approx 10+zeta(16)-sqrt(96),
(1)

which is good to 6 digits.

M. Hudson (pers. comm., Nov. 8, 2004) found the approximations

zeta(3) approx 8-ln896
(2)
 approx (pi+pi^2)^(69/962)
(3)
 approx ((71)/(47)+gamma)^(1/4)
(4)
 approx phi-(1307)/(3142)
(5)
 approx ((19963)/(4265)+phi)^(1/10)
(6)
 approx 525587^(5123^(-1/2)),
(7)

where gamma is the Euler-Mascheroni constant and phi is the golden ratio, which are good to 5, 7, 7, 8, 11, and 12 digits, respectively.

A curious approximation to zeta(3) is given by

 zeta(3) approx gamma^(-1/3)+pi^(-4)(1+2gamma-2/(130+pi^2))^(-3),
(8)

where gamma is the Euler-Mascheroni constant, which is accurate to four digits (P. Galliani, pers. comm., April 19, 2002).

Lima (2009) found the approximation

 zeta(3) approx -5/(197)+(11)/(394)pi^2ln2-(283)/(394)piln^22+(235)/(197)ln^32+(209)/(394)ln^3(1+sqrt(2))+(93)/(197)piK,
(9)

where K is Catalan's constant, which is correct to 21 digits.


See also

Almost Integer, Apéry's Constant

Explore with Wolfram|Alpha

References

Lima, F. M. S. "A Simple Approximate Expression for the ApŽry's Constant Accurate to 21 Digits." http://arxiv.org/abs/0910.2684/. 14 Oct 2009.

Referenced on Wolfram|Alpha

Apéry's Constant Approximations

Cite this as:

Weisstein, Eric W. "Apéry's Constant Approximations." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/AperysConstantApproximations.html

Subject classifications