Apéry's Constant Approximations

Approximations to Apéry's constant include

(1)

which is good to 6 digits (E. Pegg Jr., pers. comm., Nov. 8, 2004).

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

			(2)
			(3)
			(4)
			(5)
			(6)
			(7)

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

E. Weisstein (using Wolfram|Alpha, Oct. 29, 2026) obtained the approximations

			(8)
			(9)

which are good to 8 and 14 decimal digits, respectively.

A curious approximation to is given by

(10)

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

Lima (2009) found the approximation

(11)

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

T. Rialan and A. Kirilov (pers. comm., June 16, 2024) found the approximation

(12)

The absolute error is approximately .