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Apéry's Constant Digits

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Apéry's constant is defined by

zeta(3)=1.2020569...,
(1)

(OEIS A002117) where zeta(z) is the Riemann zeta function.

B. Haible and T. Papanikolaou computed zeta(3) to 1000000 digits using a Wilf-Zeilberger pair identity with

F(n,k)=(-1)^k((n!)^6(2n-k-1)!k!^3)/(2(n+k+1)!^2(2n)!^3),
(2)

s=1, and t=1, giving the rapidly converging

zeta(3)=sum_(n=0)^infty(-1)^n((n!)^(10)(205n^2+250n+77))/(64((2n+1)!)^5)
(3)

(Amdeberhan and Zeilberger 1997). The record as of Dec. 1998 was 128 million digits, computed by S. Wedeniwski. zeta(3) was computed to 10^8 decimal digits by E. Weisstein on Sep. 16, 2013.

The Earls sequence (starting position of n copies of the digit n) for zeta(3) is given for n=1, 2, ... by 10, 57, 3938, 421, 41813, 1625571, 4903435, 99713909, ... (OEIS A229074).

zeta(3)-constant prime occur for n=10, 55, 109, 141, ... (OEIS A119334), corresponding to the primes 1202056903, 1202056903159594285399738161511449990764986292340498881, ... (OEIS A119333).

The starting positions of the first occurrence of n=0, 1, 2, ... in the decimal expansion of zeta(3) (not including the initial 0 to the left of the decimal point) are 3, 1, 2, 10, 16, 6, 7, 23, 18, 8, ... (OEIS A229187).

Scanning the decimal expansion of zeta(3) until all n-digit numbers have occurred, the last 1-, 2-, ... digit numbers appearing are 7, 89, 211, 2861, 43983, 292702, 8261623, ... (OEIS A036902), which end at digits 23, 457, 7839, 83054, 1256587, 13881136, 166670757, ... (OEIS A036906).

The digit sequences 0123456789 and 9876543210 do not occur in the first 10^9 digits (E. Weisstein, Sep. 17, 2013).

It is not known if zeta(3) is normal (Bailey and Crandall 2003), but the following table giving the counts of digits in the first 10^n terms shows that the decimal digits are very uniformly distributed up to at least 10^9.

d\nOEIS1010010^310^410^510^610^710^810^9
03910899099109976110004169999248100001073
111110410241003710027310004841000016399996430
22910910071006110001210010361000557999985752
3111106101099619989499803210000695100007728
40876953995799904998174999160399994148
51131081006993310039910020431000361099999279
61790100199679952599981810003630100014221
7061131064102531006161000198999507799993290
80129098199319967599996910001192100009336
911496964999099941999830999920399998743

See also

Apéry's Constant, Apéry's Constant Continued Fraction, Constant Digit Scanning

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References

Amdeberhan, T. and Zeilberger, D. "Hypergeometric Series Acceleration via the WZ Method." Elec. J. Combin. 4, No. 2, R3, 1-3, 1997. https://doi.org/10.37236/1318.Bailey, D. H. and Crandall, R. E. "Random Generators and Normal Numbers." Exper. Math. 11, 527-546, 2002. https://doi.org/10.1080/10586458.2002.10504704.Sloane, N. J. A. Sequences A002117, A036902, A036906, A119333, A119334, A229074, and A229187 in "The On-Line Encyclopedia of Integer Sequences."Wedeniwski, S. "128000026 Digits of Zeta(3)." https://www.plouffe.fr/simon/constants/Zeta3.txt.

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Apéry's Constant Digits

Cite this as:

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

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