A258986 -id:A258986 - cp's OEIS frontend

A258987 Decimal expansion of the multiple zeta value (Euler sum) zetamult(3,3).

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

Offset: 0

Jean-François Alcover, Jun 16 2015

			0.213798868224592547099583574508033649640958957865517556144512748947...

Eric Weisstein's MathWorld, Multivariate Zeta Function
Wikipedia, Multiple zeta function
Index to constants which are multiple zeta values (3,3)

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).

RealDigits[(1/2)*Zeta[3]^2 - (1/2)*Zeta[6], 10, 103] // First (* Corrected by Detlef Meya, Jun 06 2025 *)

zetamult([3,3]) \\ Charles R Greathouse IV, Jan 21 2016

zetamult(3,3) = Sum_{m>=2} (Sum_{n=1..m-1} 1/(m^3*n^3)) = (1/2)*zeta(3)^2 - (1/2)*zeta(6). - [Corrected by Detlef Meya, Jun 06 2025 ]

A258983 Decimal expansion of the multiple zeta value (Euler sum) zetamult(3,2).

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Jun 16 2015

Also zetamult(2, 2, 1). - Charles R Greathouse IV, Jan 04 2017

			0.2288103976033537597687461489416887919325093427198821602294071...

Dominique Manchon, Arborified multiple zeta values, arXiv:1603.01498 [math.CO], 2016.
Jonathan Borwein and Roland Girgensohn, Evaluation of triple Euler Sums, Elec. Jour. of Comb., Vol. 3, Issue 1, 1996. Article R23 (see page 21).
Eric Weisstein's MathWorld, Multivariate Zeta Function
Wikipedia, Multiple zeta function

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).

Cf. A013663 (zeta(5)), A183699 (zeta(2)*zeta(3)).

RealDigits[3*Zeta[2]*Zeta[3] - (11/2)*Zeta[5], 10, 104] // First

zetamult([3,2]) \\ Charles R Greathouse IV, Jan 21 2016

zetamult([2,2,1]) \\ Charles R Greathouse IV, Jan 04 2017

Equals Sum_{m>=2} (Sum_{n=1..m-1} 1/(m^3*n^2)) = 3*zeta(2)*zeta(3) - (11/2)*zeta(5).

A258989 Decimal expansion of the multiple zeta value (Euler sum) zetamult(2,4).

Original entry on oeis.org

6, 7, 4, 5, 2, 3, 9, 1, 4, 0, 3, 3, 9, 6, 8, 1, 4, 0, 4, 9, 1, 5, 6, 0, 6, 0, 8, 2, 5, 7, 4, 2, 9, 9, 3, 9, 2, 7, 8, 3, 8, 4, 3, 6, 5, 1, 3, 7, 8, 8, 9, 5, 7, 9, 7, 0, 6, 9, 1, 7, 2, 2, 1, 4, 4, 3, 7, 7, 4, 8, 5, 8, 2, 4, 7, 7, 2, 4, 8, 5, 1, 9, 5, 6, 2, 5, 2, 6, 8, 8, 8, 5, 3, 4, 3, 0, 7, 9, 1, 2, 7, 8, 1

Offset: 0

Jean-François Alcover, Jun 16 2015

			0.67452391403396814049156060825742993927838436513788957970691722144377...

Eric Weisstein's MathWorld, Multivariate Zeta Function
Wikipedia, Multiple zeta function

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), A258990 (3,4), A258991 (4,4).

RealDigits[(25/12)*Zeta[6] - Zeta[3]^2, 10, 103] // First

zetamult([2,4]) \\ Charles R Greathouse IV, Jan 21 2016

zetamult(2,4) = Sum_{m>=2} (Sum_{n=1..m-1} 1/(m^2*n^4)) = (25/12)*zeta(6) - zeta(3)^2.

Equals Sum_{i, j >= 1} 1/(i^4*j^2*binomial(i+j, i)). - Peter Bala, Aug 05 2025

A258984 Decimal expansion of the multiple zeta value (Euler sum) zetamult(4,2).

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Jun 16 2015

			0.088483382454368714294327839085760456647978752386750591674889276559474...

Eric Weisstein's MathWorld, Multivariate Zeta Function
Wikipedia, Multiple zeta function

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).

Join[{0}, RealDigits[Zeta[3]^2 - (4/3)*Zeta[6], 10, 107] // First]

zetamult([4,2]) \\ Charles R Greathouse IV, Jan 21 2016

zetamult(4,2) = Sum_{m>=2} (Sum_{n=1..m-1} 1/(m^4*n^2)) = zeta(3)^2 - (4/3)*zeta(6).

A258985 Decimal expansion of the multiple zeta value (Euler sum) zetamult(5,2).

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Jun 16 2015

			0.03857512434275325550592546437299557001973484169890900833104937293358...

Eric Weisstein's MathWorld, Multivariate Zeta Function
Wikipedia, Multiple zeta function

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).

Join[{0}, RealDigits[5*Zeta[2]*Zeta[5] + 2*Zeta[3]*Zeta[4] - 11*Zeta[7], 10, 104] // First]

zetamult([5,2]) \\ Charles R Greathouse IV, Jan 21 2016

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).

A258988 Decimal expansion of the multiple zeta value (Euler sum) zetamult(4,3).

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Jun 16 2015

			0.0851598225348336514068060188723673459573395085868773204671034320533...

Eric Weisstein's MathWorld, Multivariate Zeta Function
Wikipedia, Multiple zeta function

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).

Join[{0}, RealDigits[17*Zeta[7] - 10*Zeta[2]*Zeta[5], 10, 104] // First]

zetamult([4,3]) \\ Charles R Greathouse IV, Jan 21 2016

zetamult(4,3) = Sum_{m>=2} (Sum_{n=1..m-1} 1/(m^2*n^3)) = 17*zeta(7) - 10*zeta(2)*zeta(5).

A258990 Decimal expansion of the multiple zeta value (Euler sum) zetamult(3,4).

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Jun 16 2015

			0.20750501461573209590780760549467146544182867955060619041951789656971...

Eric Weisstein's MathWorld, Multivariate Zeta Function
Wikipedia, Multiple zeta function

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).

RealDigits[10*Zeta[2]*Zeta[5] + Zeta[3]*Zeta[4] - 18*Zeta[7], 10, 105] // First

zetamult([3,4]) \\ Charles R Greathouse IV, Jan 21 2016

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).

A258991 Decimal expansion of the multiple zeta value (Euler sum) zetamult(4,4).

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Jun 16 2015

			0.08367311301649536161489043654238770543824673255415416836075918355438...

Eric Weisstein's MathWorld, Multivariate Zeta Function
Wikipedia, Multiple zeta function

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).

Join[{0}, RealDigits[(1/2)*(Zeta[4]^2 - Zeta[8]), 10, 106] // First]

zetamult([4,4]) \\ Charles R Greathouse IV, Jan 21 2016

(zeta(4)^2-zeta(8))/2 \\ Charles R Greathouse IV, Jan 20 2022

zetamult(4,4) = Sum_{m>=2} (Sum_{n=1..m-1} 1/(m^4*n^4)) = (1/2)*(zeta(4)^2 - zeta(8)).

A382634 Decimal expansion of the multiple prime zeta value p[2, 3].

Original entry on oeis.org

0, 2, 9, 1, 8, 5, 1, 6, 6, 5, 0, 4, 0, 1, 2, 5, 7, 1, 0, 4, 0, 6, 4, 1, 3, 4, 4, 0, 9, 0, 2, 7, 9, 1, 9, 6, 7, 4, 7

Offset: 0

Artur Jasinski, Apr 01 2025

Prime multiple zeta constants p[m,...,n] are equivalents of multiple zeta constants when successive natural numbers are replaced by successive primes.

For complete list of multiple prime zeta values up to weight 6 see A382234.

			0.029185166504012571040641344090279196747...

Cf. A085548, A085541, A085965, A382234, A382235, A382236, A382635.

Cf. A258986.

p2 = N[PrimeZetaP[2], 50]; p = 2; sum = 0; sum1 = 0; diff = 0; Monitor[Do[sum = sum + N[1/p^2, 50]; diff = p2 - sum; sum1 = sum1 + diff/p^3; p = NextPrime[p], {n, 1, 100000000}], {sum1, n}]

Equals Sum_{p,q prime p>q} 1/(p^2*q^3).
Equals A085548*A085541 - A085965 - A382635.
For partial sums and in infinity occurs identity:
p[2, 3] + p[3, 2] + p[2, 1, 2] + p[2, 2, 1] = p[1] p[2, 2]- p[1, 2, 2]
where p[1] and p[1, 2, 2] are divergent series then
lim_{n->oo} p[1](n)*A382234 - p[1, 2, 2](n) = 0.067101047034256...

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A258987 Decimal expansion of the multiple zeta value (Euler sum) zetamult(3,3).

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A258983 Decimal expansion of the multiple zeta value (Euler sum) zetamult(3,2).

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A258989 Decimal expansion of the multiple zeta value (Euler sum) zetamult(2,4).

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A258984 Decimal expansion of the multiple zeta value (Euler sum) zetamult(4,2).

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A258985 Decimal expansion of the multiple zeta value (Euler sum) zetamult(5,2).

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A258988 Decimal expansion of the multiple zeta value (Euler sum) zetamult(4,3).

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A258990 Decimal expansion of the multiple zeta value (Euler sum) zetamult(3,4).

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