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Apéry's Constant Continued Fraction

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The continued fraction for Apéry's constant zeta(3) is [1; 4, 1, 18, 1, 1, 1, 4, 1, ...] (OEIS A013631).

AperysConstantContinuedFractionFirstOccurrences

The positions at which the numbers 1, 2, ... occur in the continued fraction are 0, 11, 24, 1, 63, 26, 16, 139, 9, 118, 20, ... (OEIS A229057). The incrementally maximal terms are 1, 4, 18, 30, 428, 458, 527, ... (OEIS A033166), which occur at positions 0, 1, 3, 28, 62, 571, 1555, 2012, 2529, ... (OEIS A229055).

AperyKhinchinLevy

Let the continued fraction of zeta(3) be denoted [a_0;a_1,a_2,...] and let the denominators of the convergents be denoted q_1, q_2, ..., q_n. Then plots above show successive values of a_1^(1/1), (a_1a_2)^(1/2), (a_1a_2...a_n)^(1/n), which appear to converge to Khinchin's constant (left figure) and q_n^(1/n), which appear to converge to the Lévy constant (right figure), although neither of these limits has been rigorously established.

See also

Apéry's Constant, Apéry's Constant Digits

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References

Sloane, N. J. A. Sequences A013631, A033166, A229055, and A229057 in "The On-Line Encyclopedia of Integer Sequences."

Cite this as:

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

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