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Decimal expansion of the multiple zeta value (Euler sum) zetamult(3,3).
+10
9
2, 1, 3, 7, 9, 8, 8, 6, 8, 2, 2, 4, 5, 9, 2, 5, 4, 7, 0, 9, 9, 5, 8, 3, 5, 7, 4, 5, 0, 8, 0, 3, 3, 6, 4, 9, 6, 4, 0, 9, 5, 8, 9, 5, 7, 8, 6, 5, 5, 1, 7, 5, 5, 6, 1, 4, 4, 5, 1, 2, 7, 4, 8, 9, 4, 7, 1, 2, 5, 8, 3, 6, 6, 1, 4, 6, 9, 8, 1, 0, 2, 0, 4, 1, 7, 0, 9, 5, 6, 0, 2, 8, 9, 9, 9, 1, 1, 5, 5, 0, 6, 4, 8
FORMULA
zetamult(3,3) = Sum_{m>=2} (Sum_{n=1..m-1} 1/(m^2*n^3)) = (9/2)*zeta(5) - 2*zeta(2)*zeta(3).
EXAMPLE
0.213798868224592547099583574508033649640958957865517556144512748947...
MATHEMATICA
RealDigits[(9/2)*Zeta[5] - 2*Zeta[2]*Zeta[3], 10, 103] // First
CROSSREFS
Cf. A072691 (zetamult(1,1)), A197110 (zetamult(2,2)), A258983 (zetamult(3,2)), A258984 (4,2), A258985 (5,2), A258947 (6,2), A258986 (2,3), A258988 (4,3), A258982 (5,3), A258989 (2,4), A258990 (3,4), A258991 (4,4).
Decimal expansion of the multiple zeta value (Euler sum) zetamult(3,2).
+10
8
2, 2, 8, 8, 1, 0, 3, 9, 7, 6, 0, 3, 3, 5, 3, 7, 5, 9, 7, 6, 8, 7, 4, 6, 1, 4, 8, 9, 4, 1, 6, 8, 8, 7, 9, 1, 9, 3, 2, 5, 0, 9, 3, 4, 2, 7, 1, 9, 8, 8, 2, 1, 6, 0, 2, 2, 9, 4, 0, 7, 1, 0, 2, 6, 9, 3, 2, 2, 5, 3, 5, 8, 6, 1, 5, 2, 6, 4, 4, 5, 8, 0, 2, 6, 9, 1, 6, 0, 3, 1, 5, 0, 1, 0, 1, 5, 4, 7, 2, 0, 2, 8, 3, 7
FORMULA
zetamult(3,2) = Sum_{m>=2} (Sum_{n=1..m-1} 1/(m^3*n^2)) = 3*zeta(2)*zeta(3) - (11/2)*zeta(5).
EXAMPLE
0.2288103976033537597687461489416887919325093427198821602294071...
MATHEMATICA
RealDigits[3*Zeta[2]*Zeta[3] - (11/2)*Zeta[5], 10, 104] // First
CROSSREFS
Cf. A072691 (zetamult(1,1)), A197110 (zetamult(2,2)), A258984 (4,2), A258985 (5,2), A258947 (6,2), A258986 (2,3), A258987 (3,3), A258988 (4,3), A258982 (5,3), A258989 (2,4), A258990 (3,4), A258991 (4,4).
Decimal expansion of the multiple zeta value (Euler sum) zetamult(4,2).
+10
8
0, 8, 8, 4, 8, 3, 3, 8, 2, 4, 5, 4, 3, 6, 8, 7, 1, 4, 2, 9, 4, 3, 2, 7, 8, 3, 9, 0, 8, 5, 7, 6, 0, 4, 5, 6, 6, 4, 7, 9, 7, 8, 7, 5, 2, 3, 8, 6, 7, 5, 0, 5, 9, 1, 6, 7, 4, 8, 8, 9, 2, 7, 6, 5, 5, 9, 4, 7, 4, 2, 7, 8, 9, 2, 8, 7, 4, 3, 5, 7, 1, 4, 5, 5, 8, 2, 7, 7, 9, 4, 6, 0, 0, 4, 7, 0, 5, 8, 6, 6, 1, 9, 5, 5, 9, 6, 6, 7
FORMULA
zetamult(4,2) = Sum_{m>=2} (Sum_{n=1..m-1} 1/(m^4*n^2)) = zeta(3)^2 - (4/3)*zeta(6).
EXAMPLE
0.088483382454368714294327839085760456647978752386750591674889276559474...
MATHEMATICA
Join[{0}, RealDigits[Zeta[3]^2 - (4/3)*Zeta[6], 10, 107] // First]
CROSSREFS
Cf. A072691 (zetamult(1,1)), A197110 (zetamult(2,2)), A258983 (zetamult(3,2)), A258985 (5,2), A258947 (6,2), A258986 (2,3), A258987 (3,3), A258988 (4,3), A258982 (5,3), A258989 (2,4), A258990 (3,4), A258991 (4,4).
Decimal expansion of the multiple zeta value (Euler sum) zetamult(5,2).
+10
8
0, 3, 8, 5, 7, 5, 1, 2, 4, 3, 4, 2, 7, 5, 3, 2, 5, 5, 5, 0, 5, 9, 2, 5, 4, 6, 4, 3, 7, 2, 9, 9, 5, 5, 7, 0, 0, 1, 9, 7, 3, 4, 8, 4, 1, 6, 9, 8, 9, 0, 9, 0, 0, 8, 3, 3, 1, 0, 4, 9, 3, 7, 2, 9, 3, 3, 5, 8, 2, 3, 6, 5, 9, 1, 0, 8, 4, 5, 3, 8, 3, 6, 5, 5, 6, 8, 4, 8, 8, 2, 9, 4, 6, 4, 5, 6, 4, 7, 3, 1, 5, 5, 6, 4, 9
FORMULA
zetamult(5,2) = Sum_{m>=2} (Sum_{n=1..m-1} 1/(m^5*n^2)) = 5*zeta(2)*zeta(5) + 2*zeta(3)*zeta(4) - 11*zeta(7).
EXAMPLE
0.03857512434275325550592546437299557001973484169890900833104937293358...
MATHEMATICA
Join[{0}, RealDigits[5*Zeta[2]*Zeta[5] + 2*Zeta[3]*Zeta[4] - 11*Zeta[7], 10, 104] // First]
CROSSREFS
Cf. A072691 (zetamult(1,1)), A197110 (zetamult(2,2)), A258983 (zetamult(3,2)), A258984 (4,2), A258947 (6,2), A258986 (2,3), A258987 (3,3), A258988 (4,3), A258982 (5,3), A258989 (2,4), A258990 (3,4), A258991 (4,4).
Decimal expansion of the multiple zeta value (Euler sum) zetamult(2,3).
+10
8
7, 1, 1, 5, 6, 6, 1, 9, 7, 5, 5, 0, 5, 7, 2, 4, 3, 2, 0, 9, 6, 9, 7, 3, 8, 0, 6, 0, 8, 6, 4, 0, 2, 6, 1, 2, 0, 9, 2, 5, 6, 1, 2, 0, 4, 4, 3, 8, 3, 3, 9, 2, 3, 6, 4, 9, 2, 2, 2, 2, 4, 9, 6, 4, 5, 7, 6, 8, 6, 0, 8, 5, 7, 4, 5, 0, 5, 8, 2, 6, 5, 1, 1, 5, 4, 2, 5, 2, 3, 4, 4, 6, 3, 6, 0, 0, 7, 9, 8, 9, 6, 4, 1
FORMULA
zetamult(2,3) = Sum_{m>=2} (Sum_{n=1..m-1} 1/(m^2*n^3)) = (9/2)*zeta(5) - 2*zeta(2)*zeta(3).
EXAMPLE
0.711566197550572432096973806086402612092561204438339236492222496457686...
MATHEMATICA
RealDigits[(9/2)*Zeta[5] - 2*Zeta[2]*Zeta[3], 10, 103] // First
CROSSREFS
Cf. A072691 (zetamult(1,1)), A197110 (zetamult(2,2)), A258983 (zetamult(3,2)), A258984 (4,2), A258985 (5,2), A258947 (6,2), A258987 (3,3), A258988 (4,3), A258982 (5,3), A258989 (2,4), A258990 (3,4), A258991 (4,4).
Decimal expansion of the multiple zeta value (Euler sum) zetamult(4,3).
+10
8
0, 8, 5, 1, 5, 9, 8, 2, 2, 5, 3, 4, 8, 3, 3, 6, 5, 1, 4, 0, 6, 8, 0, 6, 0, 1, 8, 8, 7, 2, 3, 6, 7, 3, 4, 5, 9, 5, 7, 3, 3, 9, 5, 0, 8, 5, 8, 6, 8, 7, 7, 3, 2, 0, 4, 6, 7, 1, 0, 3, 4, 3, 2, 0, 5, 3, 3, 0, 8, 5, 7, 6, 7, 5, 0, 8, 7, 1, 7, 6, 6, 5, 1, 1, 1, 7, 3, 3, 8, 6, 7, 5, 8, 1, 8, 5, 0, 2, 0, 7, 2, 0, 5, 4, 1
FORMULA
zetamult(4,3) = Sum_{m>=2} (Sum_{n=1..m-1} 1/(m^2*n^3)) = 17*zeta(7) - 10*zeta(2)*zeta(5).
EXAMPLE
0.0851598225348336514068060188723673459573395085868773204671034320533...
MATHEMATICA
Join[{0}, RealDigits[17*Zeta[7] - 10*Zeta[2]*Zeta[5], 10, 104] // First]
CROSSREFS
Cf. A072691 (zetamult(1,1)), A197110 (zetamult(2,2)), A258983 (zetamult(3,2)), A258984 (4,2), A258985 (5,2), A258947 (6,2), A258986 (2,3), A258987 (3,3), A258982 (5,3), A258989 (2,4), A258990 (3,4), A258991 (4,4).
Decimal expansion of the multiple zeta value (Euler sum) zetamult(3,4).
+10
8
2, 0, 7, 5, 0, 5, 0, 1, 4, 6, 1, 5, 7, 3, 2, 0, 9, 5, 9, 0, 7, 8, 0, 7, 6, 0, 5, 4, 9, 4, 6, 7, 1, 4, 6, 5, 4, 4, 1, 8, 2, 8, 6, 7, 9, 5, 5, 0, 6, 0, 6, 1, 9, 0, 4, 1, 9, 5, 1, 7, 8, 9, 6, 5, 6, 9, 7, 1, 0, 1, 1, 9, 9, 7, 1, 6, 0, 7, 8, 0, 0, 7, 8, 0, 9, 8, 6, 6, 4, 3, 6, 3, 3, 0, 5, 2, 3, 0, 2, 0, 2, 9, 6, 5, 9
FORMULA
zetamult(3,4) = Sum_{m>=2} (Sum_{n=1..m-1} 1/(m^3*n^4)) = 10*zeta(2)*zeta(5) + zeta(3)*zeta(4) - 18*zeta(7).
EXAMPLE
0.20750501461573209590780760549467146544182867955060619041951789656971...
MATHEMATICA
RealDigits[10*Zeta[2]*Zeta[5] + Zeta[3]*Zeta[4] - 18*Zeta[7], 10, 105] // First
CROSSREFS
Cf. A072691 (zetamult(1,1)), A197110 (zetamult(2,2)), A258983 (zetamult(3,2)), A258984 (4,2), A258985 (5,2), A258947 (6,2), A258986 (2,3), A258987 (3,3), A258988 (4,3), A258982 (5,3), A258989 (2,4), A258991 (4,4).
Decimal expansion of the multiple zeta value (Euler sum) zetamult(4,4).
+10
8
0, 8, 3, 6, 7, 3, 1, 1, 3, 0, 1, 6, 4, 9, 5, 3, 6, 1, 6, 1, 4, 8, 9, 0, 4, 3, 6, 5, 4, 2, 3, 8, 7, 7, 0, 5, 4, 3, 8, 2, 4, 6, 7, 3, 2, 5, 5, 4, 1, 5, 4, 1, 6, 8, 3, 6, 0, 7, 5, 9, 1, 8, 3, 5, 5, 4, 3, 8, 1, 9, 1, 2, 7, 1, 4, 5, 6, 2, 4, 0, 1, 1, 9, 9, 6, 0, 7, 2, 6, 9, 1, 9, 7, 6, 9, 7, 6, 6, 4, 2, 6, 0, 3, 7, 6, 9, 7
FORMULA
zetamult(4,4) = Sum_{m>=2} (Sum_{n=1..m-1} 1/(m^4*n^4)) = (1/2)*(zeta(4)^2 - zeta(8)).
EXAMPLE
0.08367311301649536161489043654238770543824673255415416836075918355438...
MATHEMATICA
Join[{0}, RealDigits[(1/2)*(Zeta[4]^2 - Zeta[8]), 10, 106] // First]
CROSSREFS
Cf. A072691 (zetamult(1,1)), A197110 (zetamult(2,2)), A258983 (zetamult(3,2)), A258984 (4,2), A258985 (5,2), A258947 (6,2), A258986 (2,3), A258987 (3,3), A258988 (4,3), A258982 (5,3), A258989 (2,4), A258990 (3,4).
Decimal expansion of the infinite double sum S = Sum_{m>=1} (Sum_{n>=1} 1/(m^2*n*(m+n)^3)).
+10
0
1, 6, 9, 5, 5, 7, 1, 7, 6, 9, 9, 7, 4, 0, 8, 1, 8, 9, 9, 5, 2, 4, 1, 9, 6, 5, 4, 9, 6, 5, 1, 5, 3, 4, 2, 1, 3, 1, 6, 9, 6, 9, 5, 8, 1, 6, 7, 2, 1, 4, 2, 2, 6, 0, 3, 0, 7, 0, 6, 8, 1, 1, 0, 6, 6, 7, 3, 8, 8, 6, 9, 7, 1, 5, 0, 3, 2, 6, 3, 1, 6, 3, 1, 3, 7, 9, 5, 6, 6, 2, 9, 8, 9, 7, 5, 5, 8, 6, 1, 7, 5, 5, 0
FORMULA
S = (7/4)*zeta(6) - zeta(3)^2/2 - sum_{m>=1} (PolyGamma(1, m+1)/m^4) + (1/2)*sum_{m>=1} (PolyGamma(2, m+1)/m^3), where sum_{m>=1} (PolyGamma(1, m+1)/m^4) is A258989, the second sum being A259927. S simplifies to zeta(6)/6 = Pi^6/5670.
EXAMPLE
0.16955717699740818995241965496515342131696958167214226030706811...
MATHEMATICA
RealDigits[Pi^6/5670, 10, 103] // First
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