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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
0A0000003910899099109976110004169999248100001073
1A00000011110410241003710027310004841000016399996430
2A0000002910910071006110001210010361000557999985752
3A000000111106101099619989499803210000695100007728
4A0000000876953995799904998174999160399994148
5A0000001131081006993310039910020431000361099999279
6A0000001790100199679952599981810003630100014221
7A000000061131064102531006161000198999507799993290
8A0000000129098199319967599996910001192100009336
9A00000011496964999099941999830999920399998743

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." Electronic J. Combinatorics 4, No. 2, R3, 1-3, 1997. http://www.combinatorics.org/Volume_4/Abstracts/v4i2r3.html. Also available at http://www.math.temple.edu/~zeilberg/mamarim/mamarimhtml/accel.html.Bailey, D. H. and Crandall, R. E. "Random Generators and Normal Numbers." Exper. Math. 11, 527-546, 2002.Preprint dated Feb. 22, 2003 available at http://www.nersc.gov/~dhbailey/dhbpapers/bcnormal.pdf.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)." http://pi.lacim.uqam.ca/piDATA/Zeta3.txt.

Cite this as:

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

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