 TOPICS  # Apéry's Constant Digits

Apéry's constant is defined by (1)

(OEIS A002117) where is the Riemann zeta function.

B. Haible and T. Papanikolaou computed to digits using a Wilf-Zeilberger pair identity with (2) , and , giving the rapidly converging (3)

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

The Earls sequence (starting position of copies of the digit ) for is given for , 2, ... by 10, 57, 3938, 421, 41813, 1625571, 4903435, 99713909, ... (OEIS A229074). -constant prime occur for , 55, 109, 141, ... (OEIS A119334), corresponding to the primes 1202056903, 1202056903159594285399738161511449990764986292340498881, ... (OEIS A119333).

The starting positions of the first occurrence of , 1, 2, ... in the decimal expansion of (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 until all -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 digits (E. Weisstein, Sep. 17, 2013).

It is not known if is normal (Bailey and Crandall 2003), but the following table giving the counts of digits in the first terms shows that the decimal digits are very uniformly distributed up to at least . OEIS 10 100       0 A000000 3 9 108 990 9910 99761 1000416 9999248 100001073 1 A000000 1 11 104 1024 10037 100273 1000484 10000163 99996430 2 A000000 2 9 109 1007 10061 100012 1001036 10005579 99985752 3 A000000 1 11 106 1010 9961 99894 998032 10000695 100007728 4 A000000 0 8 76 953 9957 99904 998174 9991603 99994148 5 A000000 1 13 108 1006 9933 100399 1002043 10003610 99999279 6 A000000 1 7 90 1001 9967 99525 999818 10003630 100014221 7 A000000 0 6 113 1064 10253 100616 1000198 9995077 99993290 8 A000000 0 12 90 981 9931 99675 999969 10001192 100009336 9 A000000 1 14 96 964 9990 99941 999830 9999203 99998743

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

## Explore with Wolfram|Alpha ## 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. " 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