A258947 Decimal expansion of the multiple zeta value (Euler sum) zetamult(6,2).
0, 1, 7, 8, 1, 9, 7, 4, 0, 4, 1, 6, 8, 3, 5, 9, 8, 8, 3, 6, 2, 6, 5, 9, 5, 3, 0, 2, 4, 8, 7, 2, 4, 6, 1, 2, 1, 6, 8, 7, 1, 3, 1, 3, 7, 1, 1, 0, 2, 9, 1, 1, 8, 8, 4, 1, 8, 8, 2, 1, 3, 6, 1, 9, 1, 7, 6, 1, 3, 4, 8, 0, 2, 7, 6, 4, 1, 6, 0, 4, 6, 3, 7, 1, 8, 2, 8, 6, 2, 1, 0, 1, 9, 2, 0, 5, 8, 7, 9, 4
Offset: 0
Examples
0.01781974041683598836265953024872461216871313711029118841882136191761348...
Links
- Richard E. Crandall, Joe P. Buhler, On the evaluation of Euler Sums, Exp. Math. 3 (4) (1994) 275-285 Table 1.
- Eric Weisstein's MathWorld, Multivariate Zeta Function
- Wikipedia, Multiple zeta function
Programs
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Mathematica
digits = 99; zetamult[6,2] = NSum[HarmonicNumber[m-1, 2]/m^6, {m, 2, Infinity}, WorkingPrecision -> digits+20, NSumTerms -> 200, Method -> {"NIntegrate", "MaxRecursion" -> 18}]; Join[{0}, RealDigits[zetamult[6,2], 10, digits] // First] -
PARI
zetamult([6,2]) \\ Charles R Greathouse IV, Jan 21 2016
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PARI
zetamult([2, 2, 1, 1, 1, 1]) \\ Charles R Greathouse IV, Feb 04 2025
Formula
zetamult(6,2) = Sum_{m>=2} (sum_{n=1..m-1} 1/(m^6*n^2)).
Equals Sum_{m>=2} H(m-1, 2)/m^6, where H(n,2) is the n-th harmonic number of order 2.