# Apéry's constant

Apéry's constant is a draft programming task. It is not yet considered ready to be promoted as a complete task, for reasons that should be found in its talk page.

Apéry's constant is the sum of the reciprocals of the positive cubes.

That is, it is defined as the number ${\displaystyle \zeta (3)=\sum _{k=1}^{\infty }{\frac {1}{k^{3}}},}$ where ζ is the Riemann zeta function.

Approximately equal to:

1.2020569031595942853997381615114499907649862923404988817922715553418382057863130901864558736093352581

This constant was known and studied by many early mathematicians, but was not named until 1978, after Roger Apéry, who was first to prove it was an irrational number.

${\displaystyle \zeta (3)}$ by summing reciprocal cubes is easy to calculate, but converges very slowly. The first 1000 terms are only accurate to 6 decimal places.

There have been many fast convergence representations developed / discovered that generate correct digits much more quickly.

One of the earliest, discovered in the late 1800s by A. Markov and later widely published by Apéry is:

${\displaystyle \zeta (3)={\frac {5}{2}}\sum _{k=1}^{\infty }(-1)^{k-1}{\frac {k!^{2}}{(2k)!k^{3}}}.}$

Much better than direct calculation of ${\displaystyle \zeta (3)}$, but still only yielding about .63 correct digits per iteration.

Several even faster converging representions are available. The fastest known to date, yielding about 5.04 correct digits per term, is by Sebastian Wedeniwski.

${\displaystyle \zeta (3)={\frac {1}{24}}\sum _{k=0}^{\infty }(-1)^{k}{\frac {(2k+1)!^{3}(2k)!^{3}k!^{3}(126392k^{5}+412708k^{4}+531578k^{3}+336367k^{2}+104000k+12463)}{(3k+2)!(4k+3)!^{3}}}.}$

Task
• Show the value of Apéry's constant calculated at least three different ways.
1. Show the value of at least the first 1000 terms of ${\displaystyle \zeta (3)}$ by direct summing of reciprocal cubes, truncated to 100 decimal digits.
2. Show the value of the first 158 terms of Markov / Apéry representation truncated to 100 decimal digits.
3. Show the value of the first 20 terms of Wedeniwski representation truncated to 100 decimal digits.

See also

## ALGOL 68

Works with: ALGOL 68G version Any - tested with release 2.8.3.win32

Uses Algol 68G's LONG LONG INT and LONG LONG REAL which have programmer specified precision. The factorials can get quuite large so more than 101 digits are needed.

BEGIN # find Apéry's constant: the sum of the positive cubes' reciprocals     #
# (this is the value of the Riemann zeta function applied to 3)         #

PR precision 1000 PR                    # set precision of LONG LONG REAL #

# returns a string representation of z, truncated to 100 decimals         #
PROC truncate100 = ( LONG LONG REAL z )STRING:
BEGIN
STRING result = fixed( z, 0, 101 );  # format with 101 decimals  #
result[ : UPB result - 1 ]             # remove the final digit  #
END # truncate100 # ;

# methind 1 - sum the reciprocols of the cubes - 1000 terms from 1        #
BEGIN
LONG LONG REAL zeta3 := 0;
FOR k TO 1000 DO
LONG LONG INT llk = LENG LENG k;
zeta3 +:= 1 / ( llk * llk * llk )
OD;
print( ( truncate100( zeta3 ), newline ) )
END;

# method 2 - Markov's alternative representaton, 158 terms from 1         #
#     5/2 * sum [ (-1)^(k-1)( k!^2 / (2k!)k^2 ) ] from 1                  #
BEGIN
LONG LONG INT fk := 1, f2k := 1;
LONG LONG REAL zeta3 := 0;
FOR k TO 158 DO
LONG LONG INT llk  = k;
LONG LONG INT ll2k = llk * 2;
fk  *:= llk;
f2k *:= ( ll2k - 1 ) *:= ll2k;
LONG LONG REAL term = ( fk * fk ) / ( f2k * llk * llk * llk );
IF ODD k THEN
zeta3 +:= term
ELSE
zeta3 -:= term
FI
OD;
zeta3 *:= 5 / 2;
print( ( truncate100( zeta3 ), newline ) )
END;

# method 3 - Wedeniwski representation - 20 terms from 0                  #
#    1/24 * sum [ (-1)^k (2k+1)!^3(2k)!^3(k!)^3                           #
#               * ( 126392k^5 + 412708k^4 + 531578k^3                     #
#                 + 336367k^2 + 104000k   + 12463                         #
#                 ) / (3k+2)! (4k + 3)!^3                                 #
#               ] from 0                                                  #
BEGIN
[]INT w coefficients = ( 126392, 412708, 531578, 336367, 104000, 12463 );
LONG LONG INT fk := 1, f2k := 1, f3k := 1, f4k := 1;
# ensure the divisor is a LONG LONG INT so the LHS is calculated as a #
# LONG LONG REAL value and not a REAL which is then widened           #
LONG LONG REAL zeta3 := w coefficients[ UPB w coefficients ]
/ LENG LENG ( 2 * 6 * 6 * 6 );
FOR k TO 19 DO
LONG LONG INT llk  = k;
LONG LONG INT ll2k = llk  + llk;
LONG LONG INT ll3k = ll2k + llk;
LONG LONG INT ll4k = ll3k + llk;
fk  *:= llk;
f2k *:=                               ( ll2k - 1 ) * ll2k;
f3k *:=                ( ll3k - 2 ) * ( ll3k - 1 ) * ll3k;
f4k *:= ( ll4k - 3 ) * ( ll4k - 2 ) * ( ll4k - 1 ) * ll4k;
LONG LONG INT f2k1  = f2k * ( ll2k + 1 );
LONG LONG INT f3k2  = f3k * ( ll3k + 1 ) * ( ll3k + 2 );
LONG LONG INT f4k3  = f4k * ( ll4k + 1 ) * ( ll4k + 2 ) * ( ll4k + 3 );
LONG LONG REAL term := 0;
FOR c TO UPB w coefficients DO
term *:= llk +:= w coefficients[ c ]
OD;
LONG LONG INT fp = f2k1 * f2k * fk;
term *:= fp * fp * fp /:= f3k2 * f4k3 * f4k3 * f4k3;
IF ODD k THEN
zeta3 -:= term
ELSE
zeta3 +:= term
FI
OD;
zeta3 /:= 24;
print( ( truncate100( zeta3 ), newline ) )
END

END
Output:
1.2020564036593442854830714115115999903483212709031775135036540966118572571921400836130084123260473111
1.2020569031595942853997381615114499907649862923404988817922715553418382057863130901864558736093352581
1.2020569031595942853997381615114499907649862923404988817922715553418382057863130901864558736093352581


## F#

// Apéry's constant. Nigel Galloway: March 3rd., 2023
open MathNet.Numerics
let fact=let g=Seq.unfold(fun(n,g)->Some(n,(n*g,g+1N)))(1N,2N)|>Seq.cache in (fun n->Seq.item (n-1) g)
let fN g=let g=BigRational.FromInt g in 126392N*g**5+412708N*g**4+531578N*g**3+336367N*g**2+104000N*g+12463N
let fG n g l=let i=n/g in (int i,Seq.unfold(fun(n,i)->if i=0 then None else let l=n/g in Some(int l,(10I*(n-l*g),i-1)))(10I*(n-i*g),l))
let r3=Seq.initInfinite(fun g->BigRational.PowN(((+)1>>BigRational.FromInt>>BigRational.Reciprocal)g,3))|>Seq.take 1000|>Seq.sum
let ma=(5N/2N)*(Seq.unfold(fun(n,g,l)->Some(n*g,(-n,(fact l)*(fact l)/(fact(2*l)*BigRational.FromInt(pown l 3)),l+1)))(1N,1N/2N,2)|>Seq.take 158|>Seq.sum)
let sw=(1N/24N)*(Seq.unfold(fun(n,g,l)->Some(n*g,(-n,(fact(2*l+1)**3*fact(2*l)**3*(fact l)**3*(fN l))/(fact(3*l+2)*(fact(4*l+3)**3)),l+1)))(1N,12463N/432N,1)|>Seq.take 20|>Seq.sum)
[r3;ma;sw]|>List.iter(fun n->let n,g=fG (n.Numerator) (n.Denominator) 100 in printf $"%d{n}."; g|>Seq.iter(printf "%d"); printfn "")  Output: 1.2020564036593442854830714115115999903483212709031775135036540966118572571921400836130084123260473111 1.2020569031595942853997381615114499907649862923404988817922715553418382057863130901864558736093352581 1.2020569031595942853997381615114499907649862923404988817922715553418382057863130901864558736093352581  ### Bonus 4th. way as a continued fraction I extend Continued fraction#Apéry's_constant to provide the 101 digits required here. This method requires 31 iterations to provide the 101 digits, not quite as good as Wedeniwski but much better than the other 2. This method has historic interest as Apéry used it to prove that this number is irrational. let cf2br α β=let n0,g1,n1,g2=β(),α(),β(),β() seq{let (Π:BigRational)=g1/n1 in yield n0+Π; yield! Seq.unfold(fun(n,g,Π)->let a,b=α(),β() in let Π=Π*g/n in Some(n0+Π,(b+a/n,b+a/g,Π)))(g2+α()/n1,g2,Π)} let aπ()=let mutable n=0N in (fun ()->n<-n+1N; -(BigRational.Pow(n,6))) let bπ()=let mutable n=0N in (fun ()->n<-n+1N; (2N*n-1N)*(17N*n*n-17N*n+5N)) cf2br (aπ()) (bπ())|>Seq.skip 31|>Seq.take 1|>Seq.iter(fun n->let n=6N/n in let n,g=fG (n.Numerator) (n.Denominator) 100 in printf$"%d{n}."; g|>Seq.iter(printf "%d"); printfn "")

Output:
1.2020569031595942853997381615114499907649862923404988817922715553418382057863130901864558736093352581


## J

app3=: {{ +/_3x^~1+i.y }}
app3m=: {{ _5r2 * +/ {{ (_1^y)*(2^~!y)%(!2*y)*y^3}} 1x+i.y }}
app3sm=: {{ 1r24* +/ {{
(_1^y)*(3^~*/!y,0 1+/2*y)*(12463 104000 336367 531578 412708 126392 p. y)%(!2 3 p.y)*(!3 4 p.y)^3
}} x:i.y
}}

0j100 ": app3 1000
1.2020564036593442854830714115115999903483212709031775135036540966118572571921400836130084123260473112
0j100 ": app3m 158
1.2020569031595942853997381615114499907649862923404988817922715553418382057863130901864558736093352581
0j100 ": app3sm 20
1.2020569031595942853997381615114499907649862923404988817922715553418382057863130901864558736093352581


## Julia

using SpecialFunctions

setprecision(120, base=10)

println("Apéry's constant via Julia's zeta:\n$(string(zeta(big"3"))[1:102])") """ zeta(3) via Riemann summation of 1/(k cubed) """ Apéry_r(nterms = 1_000_000) = sum(big"1" / k^big"3" for k in 1:nterms) println("\nApéry's constant via reciprocal cubes:\n$(string(Apéry_r())[1:102])")

""" zeta(3) via Markov's summation """
function Apéry_m(nterms = 158)
return big"2.5" * sum((isodd(k) ? 1 : -1) * factorial(big(k))^2 /
(factorial(big"2" * k) * k^big"3") for k in 1:nterms)
end

println("\nApéry's constant via Markov's summation:\n$(string(Apéry_m())[1:102])") """ zeta(3) via Wedeniwski's summation """ function Apéry_w(nterms = 20) return big"1"/24 * sum((iseven(k) ? 1 : -1) * factorial(big"2" * k + 1)^3 * factorial(big"2" * k)^3 * factorial(big(k))^3 * (126392 * k^big"5" + 412708 * k^big"4" + 531578 * k^big"3" + 336367 * k^big"2" + big"104000" * k + 12463) / (factorial(big"3" * k + 2) * factorial(big"4" * k+3)^3) for k in 0:nterms) end println("\nApéry's constant via Wedeniwski's summation:\n$(string(Apéry_w())[1:102])")

Output:
Apéry's constant via Julia's zeta:
1.2020569031595942853997381615114499907649862923404988817922715553418382057863130901864558736093352581

Apéry's constant via reciprocal cubes:
1.2020569031590942858997379115114499908483196256737488817922717053418382053696464235214344450378979367

Apéry's constant via Markov's summation:
1.2020569031595942853997381615114499907649862923404988817922715553418382057863130901864558736093352581

Apéry's constant via Wedeniwski's summation:
1.2020569031595942853997381615114499907649862923404988817922715553418382057863130901864558736093352581


## Mathematica/Wolfram Language

ClearAll["Global*"];
TruancateTo100DecimalDigits = N[#, 100 + 1] &;
MyShowApéryConstant[expr_, caption_String] :=
Print[caption <>
ToString@Activate@TruancateTo100DecimalDigits[expr]];
MyShowApéryConstant[
Zeta[3], "Apéry's constant via Mathematica's Zeta:\n"]
MyShowApéryConstant[
Sum[1/(k^3), {k, 1,
1000}], "Apéry's constant via reciprocal cubes:\n"]
MyShowApéryConstant[(5/2*
Sum[(-1)^(k - 1)*(k!)^2/((2 k)!*k^3), {k, 1,
158}]), "Apéry's constant via Markov's summation:\n"]
MyShowApéryConstant[
1/24*Sum[(-1)^
k*((2 k + 1)!)^3*((2 k)!)^3*(k!)^3*(126392 k^5 + 412708 k^4 +
531578 k^3 + 336367 k^2 + 104000 k +
12463)/(((3 k + 2)!)*((4 k + 3)!)^3), {k, 0,
19}], "Apéry's constant via Wedeniwski's summation:\n"]

Output:
Apéry's constant via Mathematica's Zeta:

1.2020569031595942853997381615114499907649862923404988817922715553418382057863130901864558736093352581

Apéry's constant via reciprocal cubes (accurate to 6 decimal places):

1.2020564036593442854830714115115999903483212709031775135036540966118572571921400836130084123260473112

Apéry's constant via Markov's summation: 1.2020569031595942853997381615114499907649862923404988817922715553418382057863130901864558736093352581

Apéry's constant via Wedeniwski's summation: 1.2020569031595942853997381615114499907649862923404988817922715553418382057863130901864558736093352581

## Nim

Translation of: Wren
Library: bignum
import std/strformat
import bignum

func toDecimal100(r: Rat): string =
## Return the representation of a rational up to 100 decimals.
r *= newInt(10)^100
result.setLen(102)
result = ($r.toInt)[0..100] result.insert(".", 1) proc apery(n: Positive) = var sum = newRat() for k in 1..n: sum += newRat(1, k^3) echo &"First {n} terms of ζ(3) truncated to 100 decimal places (accurate to 6 decimal places):" echo sum.toDecimal100 echo() proc markov(n: Positive) = var neg = true var fact1, fact2 = newInt(1) var sum = newRat() for k in 1..n: neg = not neg fact1 *= k var num = fact1 * fact1 if neg: num = -num fact2 *= 2 * k * (2 * k - 1) let denom = fact2 * k^3 sum += newRat(num, denom) sum *= newRat(5, 2) echo &"First {n} terms of Markov / Apéry representation truncated to 100 decimal places:" echo sum.toDecimal100 echo() proc wedeniwski(n: Positive) = var fact1, fact2 = newInt(1) var neg = true var sum = newRat() for k in 0..<n: neg = not neg if k > 0: fact1 *= k fact2 *= 2 * k * (2 * k - 1) let fact3 = fact2 * (2 * k + 1) var num = (fact1 * fact2 * fact3)^3 num *= ((((126392 * k + 412708) * k + 531578) * k + 336367) * k + 104000) * k + 12463 if neg: num = -num let denom = fac(4 * k + 3)^3 * fac(3 * k + 2) sum += newRat(num, denom) sum /= 24 echo &"First {n} terms of Wedeniwski representation truncated to 100 decimal places:" echo sum.toDecimal100 echo() echo "Actual value to 100 decimal places:" echo "1.2020569031595942853997381615114499907649862923404988817922715553418382057863130901864558736093352581" echo() apery(1000) markov(158) wedeniwski(20)  Output: Actual value to 100 decimal places: 1.2020569031595942853997381615114499907649862923404988817922715553418382057863130901864558736093352581 First 1000 terms of ζ(3) truncated to 100 decimal places (accurate to 6 decimal places): 1.2020564036593442854830714115115999903483212709031775135036540966118572571921400836130084123260473111 First 158 terms of Markov / Apéry representation truncated to 100 decimal places: 1.2020569031595942853997381615114499907649862923404988817922715553418382057863130901864558736093352581 First 20 terms of Wedeniwski representation truncated to 100 decimal places: 1.2020569031595942853997381615114499907649862923404988817922715553418382057863130901864558736093352581  ## PARI/GP \\ Set the precision to, say, 110 digits default(realprecision, 110); \\ Function to display Apéry's constant my_show_apery_constant(expr, caption) = printf("%s %.101g\n\n", caption, expr); \\ Apéry's constant via PARI/GP's zeta function my_show_apery_constant(zeta(3), "Apéry's constant via PARI/GP's Zeta:\n"); \\ Apéry's constant via reciprocal cubes my_show_apery_constant(sum(k = 1, 1000, 1/k^3), "Apéry's constant via reciprocal cubes:\n"); \\ Apéry's constant via Markov's summation my_show_apery_constant((5/2) * sum(k = 1, 158, (-1)^(k - 1) * (k!)^2 / ((2*k)! * k^3)), "Apéry's constant via Markov's summation:\n"); \\ Apéry's constant via Wedeniwski's summation my_show_apery_constant(1/24 * sum(k = 0, 19, (-1)^k * ((2*k + 1)!)^3 * ((2*k)!)^3 * (k!)^3 * (126392*k^5 + 412708*k^4 + 531578*k^3 + 336367*k^2 + 104000*k + 12463) / (((3*k + 2)!) * ((4*k + 3)!)^3)), "Apéry's constant via Wedeniwski's summation:\n"); Output: Apéry's constant via PARI/GP's Zeta: 1.2020569031595942853997381615114499907649862923404988817922715553418382057863130901864558736093352581 Apéry's constant via reciprocal cubes: 1.2020564036593442854830714115115999903483212709031775135036540966118572571921400836130084123260473112 Apéry's constant via Markov's summation: 1.2020569031595942853997381615114499907649862923404988817922715553418382057863130901864558736093352581 Apéry's constant via Wedeniwski's summation: 1.2020569031595942853997381615114499907649862923404988817922715553418382057863130901864558736093352581  ## Perl use v5.36; use bigrat try => 'GMP'; sub f { my$r = 1; $r *=$_ for 1..shift; $r } say 'Actual value to 100 decimal places:'; say '1.2020569031595942853997381615114499907649862923404988817922715553418382057863130901864558736093352581'; say "\nFirst 1000 terms of ζ(3) truncated to 100 decimal places. (accurate to 6 decimal places):"; my$z3;
$z3 += 1/$_**3 for 1..1000;
say $z3->as_float(101); say "\nFirst 158 terms of Markov / Apéry representation truncated to 100 decimal places:";$z3 = 0;
$z3 += (-1)**($_-1) * (f($_)**2 / (f(2*$_) * $_**3)) for 1..158;$z3 *= 5/2;
say $z3->as_float(101); say "\nFirst 20 terms of Wedeniwski representation truncated to 100 decimal places:";$z3 = 0;
$z3 += (-1)**$_ *  f(2*$_+1)**3 * f(2*$_)**3 * f($_)**3 * (126392*$_**5 + 412708*$_**4 + 531578*$_**3 + 336367*$_**2 + 104000*$_ + 12463)
/ ( f(3*$_+2) * f(4*$_+3)**3 )
for 0..19;
$z3 *= 1/24; say$z3->as_float(101);

Output:
Actual value to 100 decimal places:
1.2020569031595942853997381615114499907649862923404988817922715553418382057863130901864558736093352581

First 1000 terms of ζ(3) truncated to 100 decimal places. (accurate to 6 decimal places):
1.2020564036593442854830714115115999903483212709031775135036540966118572571921400836130084123260473112

First 158 terms of Markov / Apéry representation truncated to 100 decimal places:
1.2020569031595942853997381615114499907649862923404988817922715553418382057863130901864558736093352581

First 20 terms of Wedeniwski representation truncated to 100 decimal places:
1.2020569031595942853997381615114499907649862923404988817922715553418382057863130901864558736093352581

## Phix

Ugh. If you ran this on the James Webb, you might just be able to pick out a faint small print outline of the word "elegant".
Still, at least it is not like you do this sort of thing every day... and I got to fix a couple of bugs in my mpfr.js code.

with javascript_semantics
requires("1.0.2") -- (missing mpfr_ui_pow_ui() and bug in mpfr_mul_d(), both in mpfr.js)
include mpfr.e
mpfr_set_default_precision(-100)
mpfr {d,a,w,t} = mpfr_inits(4)
mpz {z,pk} = mpz_inits(2)
for k=1 to 1000 do
mpfr_ui_pow_ui(t,k,3)
mpfr_si_div(t,1,t)
mpfr_add(d,d,t)
end for

for k=1 to 158 do
mpz_fac_ui(z,k)
mpz_mul(z,z,z)
mpfr_set_z(t,z)
mpz_fac_ui(z,2*k)
mpfr_div_z(t,t,z)
mpz_ui_pow_ui(z,k,3)
mpfr_div_z(t,t,z)
if even(k) then
mpfr_sub(a,a,t)
else
mpfr_add(a,a,t)
end if
end for
mpfr_mul_d(a,a,5/2)

for k=0 to 19 do
mpz_ui_pow_ui(z,k,5)
mpz_mul_si(z,z,126392)
mpz_ui_pow_ui(pk,k,4)
mpz_mul_si(pk,pk,412708)
mpz_add(z,z,pk)
mpz_ui_pow_ui(pk,k,3)
mpz_mul_si(pk,pk,531578)
mpz_add(z,z,pk)
mpz_add_si(z,z,k*k*336367)
mpz_add_si(z,z,k*104000)
mpz_add_si(z,z,12463)
mpfr_set_z(t,z)
mpz_fac_ui(z,2*k+1)
mpz_pow_ui(z,z,3)
mpfr_mul_z(t,t,z)
mpz_fac_ui(z,2*k)
mpz_pow_ui(z,z,3)
mpfr_mul_z(t,t,z)
mpz_fac_ui(z,k)
mpz_pow_ui(z,z,3)
mpfr_mul_z(t,t,z)
mpz_fac_ui(z,3*k+2)
mpfr_div_z(t,t,z)
mpz_fac_ui(z,4*k+3)
mpz_pow_ui(z,z,3)
mpfr_div_z(t,t,z)
if odd(k) then
mpfr_sub(w,w,t)
else
mpfr_add(w,w,t)
end if
end for
mpfr_div_si(w,w,24)

constant fmt = """
Actual value to 100 decimal places:
1.2020569031595942853997381615114499907649862923404988817922715553418382057863130901864558736093352581

First 1000 terms of zeta(3) truncated to 100 decimal places. (accurate to 6 decimal places):
%s

First 158 terms of Markov / Apery representation truncated to 100 decimal places:
%s

First 20 terms of Wedeniwski representation truncated to 100 decimal places:
%s

"""
string direct = mpfr_get_fixed(d,100),
mapery = mpfr_get_fixed(a,100),
wdnski = mpfr_get_fixed(w,100)
printf(1,fmt,{direct,mapery,wdnski})

Output:
Actual value to 100 decimal places:
1.2020569031595942853997381615114499907649862923404988817922715553418382057863130901864558736093352581

First 1000 terms of zeta(3) truncated to 100 decimal places. (accurate to 6 decimal places):
1.2020564036593442854830714115115999903483212709031775135036540966118572571921400836130084123260473111

First 158 terms of Markov / Apery representation truncated to 100 decimal places:
1.2020569031595942853997381615114499907649862923404988817922715553418382057863130901864558736093352581

First 20 terms of Wedeniwski representation truncated to 100 decimal places:
1.2020569031595942853997381615114499907649862923404988817922715553418382057863130901864558736093352581


Last digit of the 1000 terms line is 2 under pwa/p2js...
As per Wren, you can verify or completely replace all this with mpfr_zeta_ui(w,3) [on desktop/Phix only, not supported under pwa/p2js]

## Python

from sympy import zeta, factorial
from decimal import Decimal, getcontext

# Set the desired precision
getcontext().prec = 120

def my_sympy_format_to_decimal(sympy_result):
return Decimal(str(sympy_result.evalf(getcontext().prec)))

def print_apery_constant(description, value):
print(f"{description}:\n{str(value)[:102]}")

# Apéry's constant via SymPy's zeta function
zeta_3_str = str(zeta(3).evalf(getcontext().prec))
zeta_3_decimal = Decimal(zeta_3_str)
print_apery_constant("Apéry's constant via SymPy's zeta", zeta_3_decimal)

# Apéry's constant via Riemann summation of 1/(k cubed)
def apery_r(nterms=1_000):
total = sum(Decimal('1') / Decimal(k) ** 3 for k in range(1, nterms + 1))
return total
print_apery_constant("Apéry's constant via reciprocal cubes", apery_r())

# Apéry's constant via Markov's summation
def apery_m(nterms=158):
total = Decimal(2.5) * sum(
(Decimal(1) if k % 2 != 0 else Decimal(-1)) *
my_sympy_format_to_decimal(factorial(k) ** 2) /
my_sympy_format_to_decimal(factorial(2*k)  * (k ** 3) )
for k in range(1, nterms + 1)
)
return total
print_apery_constant("Apéry's constant via Markov's summation", apery_m())

# Apéry's constant via Wedeniwski's summation
def apery_w(nterms=20):
total = Decimal('1') / Decimal('24') * sum(
(Decimal('1') if k % 2 == 0 else Decimal('-1')) *
my_sympy_format_to_decimal(factorial(2 * k + 1)) ** 3 *
my_sympy_format_to_decimal(factorial(2 * k)) ** 3 *
my_sympy_format_to_decimal(factorial(k)) ** 3 *
(Decimal('126392') * Decimal(k) ** 5 +
Decimal('412708') * Decimal(k) ** 4 +
Decimal('531578') * Decimal(k) ** 3 +
Decimal('336367') * Decimal(k) ** 2 +
Decimal('104000') * Decimal(k) +
Decimal('12463')) /
(my_sympy_format_to_decimal(factorial(3 * k + 2)) * my_sympy_format_to_decimal(factorial(4 * k + 3)) ** 3)
for k in range(0, nterms + 1)
)
return total
print_apery_constant("Apéry's constant via Wedeniwski's summation", apery_w())

Output:
Apéry's constant via SymPy's zeta:
1.2020569031595942853997381615114499907649862923404988817922715553418382057863130901864558736093352581
Apéry's constant via reciprocal cubes:
1.2020564036593442854830714115115999903483212709031775135036540966118572571921400836130084123260473111
Apéry's constant via Markov's summation:
1.2020569031595942853997381615114499907649862923404988817922715553418382057863130901864558736093352581
Apéry's constant via Wedeniwski's summation:
1.2020569031595942853997381615114499907649862923404988817922715553418382057863130901864558736093352581


## Raku

sub postfix:<!> (Int $n) { (constant f = 1, |[\×] 1..*)[$n] }

say 'Actual value to 100 decimal places:';
say '1.2020569031595942853997381615114499907649862923404988817922715553418382057863130901864558736093352581';

say "\nFirst 1000 terms of ζ(3) truncated to 100 decimal places. (accurate to 6 decimal places):";
say (1..1000).map({FatRat.new: 1, .³}).sum.substr: 0, 102;

say "\nFirst 158 terms of Markov / Apéry representation truncated to 100 decimal places:";
say (5/2 × (1..158).map( -> \k { (-1)**(k-1) × FatRat.new: k!², ((2×k)! × k³) } ).sum).substr: 0, 102;

say "\nFirst 20 terms of Wedeniwski representation truncated to 100 decimal places:";
say (1/24 × ((^20).map: -> \k {
(-1)**k × FatRat.new: (2×k+1)!³ × (2×k)!³ × k!³ × (126392×k⁵ + 412708×k⁴ + 531578×k³ + 336367×k² + 104000×k + 12463), (3×k+2)! × (4×k+3)!³
}).sum).substr: 0, 102;

Output:
Actual value to 100 decimal places:

1.2020569031595942853997381615114499907649862923404988817922715553418382057863130901864558736093352581

First 1000 terms of ζ(3) truncated to 100 decimal places. (accurate to 6 decimal places): 1.2020564036593442854830714115115999903483212709031775135036540966118572571921400836130084123260473111

First 158 terms of Markov / Apéry representation truncated to 100 decimal places: 1.2020569031595942853997381615114499907649862923404988817922715553418382057863130901864558736093352581

First 20 terms of Wedeniwski representation truncated to 100 decimal places: 1.2020569031595942853997381615114499907649862923404988817922715553418382057863130901864558736093352581

## Sidef

local Num!PREC = 4*101
say "Actual value to 100 decimal places:\n#{zeta(3)}"

say "\nFirst 1000 terms of ζ(3) truncated to 100 decimal places. (accurate to 6 decimal places):";
say sum(1..1000, {|k| 1/k**3 }).as_float(101)

say "\nFirst 158 terms of Markov / Apéry representation truncated to 100 decimal places:";
say ((5/2)*sum(1..158, {|k|
(-1)**(k-1) * (k!**2 / ((2*k)! * k**3))
}) -> as_float(101))

say "\nFirst 20 terms of Wedeniwski representation truncated to 100 decimal places:";
say ((1/24)*sum(^20, {|k|
(-1)**k * (2*k + 1)!**3 * (2*k)!**3 * k!**3 * (
126392*k**5 + 412708*k**4 + 531578*k**3 + 336367*k**2 + 104000*k + 12463
) / ((3*k + 2)! * (4*k + 3)!**3)
}) -> as_float(101))

Output:
Actual value to 100 decimal places:
1.2020569031595942853997381615114499907649862923404988817922715553418382057863130901864558736093352581

First 1000 terms of ζ(3) truncated to 100 decimal places. (accurate to 6 decimal places):
1.2020564036593442854830714115115999903483212709031775135036540966118572571921400836130084123260473112

First 158 terms of Markov / Apéry representation truncated to 100 decimal places:
1.2020569031595942853997381615114499907649862923404988817922715553418382057863130901864558736093352581

First 20 terms of Wedeniwski representation truncated to 100 decimal places:
1.2020569031595942853997381615114499907649862923404988817922715553418382057863130901864558736093352581


## Wren

Library: Wren-big
import "./big" for BigInt, BigRat

var apery = Fn.new { |n|
var sum = BigRat.zero
for (k in 1..n) sum = sum + BigRat.new(1, k*k*k)
System.print("First %(n) terms of ζ(3) truncated to 100 decimal places (accurate to 6 decimal places):")
System.print(sum.toDecimal(100, false))
System.print()
}

var markov = Fn.new { |n|
var fact = BigInt.one
var fact2 = BigInt.one
var sign = BigInt.minusOne
var sum = BigRat.zero
for (k in 1..n) {
sign = sign * BigInt.minusOne
fact = fact * k
var num = fact.square * sign
var mult = 2 * k * (2*k - 1)
fact2 = fact2 * mult
var cube = k * k * k
var den = fact2 * cube
sum = sum + BigRat.new(num, den)
}
sum = sum * BigRat.new(5, 2)
System.print("First %(n) terms of Markov / Apéry representation truncated to 100 decimal places:")
System.print(sum.toDecimal(100, false))
System.print()
}

var wedeniwski = Fn.new { |n|
var fact = BigInt.one
var fact2 = BigInt.one
var sign = BigInt.minusOne
var sum = BigRat.zero
var mult = 1
for (k in 0..n-1) {
sign = sign * BigInt.minusOne
if (k > 0) {
fact = fact * k
mult = 2 * k * (2*k - 1)
fact2 = fact2 * mult
}
var fact3 = fact2 * (2*k + 1)
var num = sign * fact3.cube * fact2.cube * fact.cube
var cube = k * k * k
var quad = cube * k
var pent = quad * k
var tmp =  126392*pent + 412708*quad + 531578*cube + 336367*k*k + 104000*k + 12463
num = num * tmp
var den = BigInt.factorial(3*k + 2) * BigInt.factorial(4*k + 3).cube
sum = sum + BigRat.new(num, den)
}
sum = sum / 24
System.print("First %(n) terms of Wedeniwski representation truncated to 100 decimal places:")
System.print(sum.toDecimal(100, false))
System.print()
}

System.print("Actual value to 100 decimal places:")
System.print("1.2020569031595942853997381615114499907649862923404988817922715553418382057863130901864558736093352581")
System.print()
apery.call(1000)
markov.call(158)
wedeniwski.call(20)

Output:
Actual value to 100 decimal places:
1.2020569031595942853997381615114499907649862923404988817922715553418382057863130901864558736093352581

First 1000 terms of ζ(3) truncated to 100 decimal places (accurate to 6 decimal places):
1.2020564036593442854830714115115999903483212709031775135036540966118572571921400836130084123260473111

First 158 terms of Markov / Apéry representation truncated to 100 decimal places:
1.2020569031595942853997381615114499907649862923404988817922715553418382057863130901864558736093352581

First 20 terms of Wedeniwski representation truncated to 100 decimal places:
1.2020569031595942853997381615114499907649862923404988817922715553418382057863130901864558736093352581


We can also verify the actual value of Apéry's constant to 100 decimal places using MPFR which has a zeta function built in. A precision of 324 bits is needed.

Library: Wren-gmp
import "./gmp" for Mpf

var x = Mpf.new(324)
var zeta = x.zetaUi(3)
System.print(zeta.toString(101))

Output:
1.2020569031595942853997381615114499907649862923404988817922715553418382057863130901864558736093352581
`