A129663 Denominators of the Pierce partial sums for L(3, chi3), where L(s, chi3) is the Dirichlet L-function for the non-principal character modulo 3.

1, 1, 8, 26, 1664, 106496, 370126848, 7279690096640, 4045738169062195200, 597704977138451388530688000, 111845949979901797334235660288000, 1194765595895193218918930427630975811584000

Offset: 0

Author

Stuart Clary, Apr 30 2007

Keywords

• nonn
• frac
• easy

Examples

			L(3, chi3) = 0.8840238117500798567430579168710118077... = 1/1 - 1/(1*8) + 1/(1*8*13) - 1/(1*8*13*16) + 1/(1*8*13*16*64) - ..., the partial sums of which are 0, 1, 7/8, 23/26, 1471/1664, 94145/106496, ...
		

References

• Leonhard Euler, "Introductio in Analysin Infinitorum", First Part, Articles 176 and 292

Crossrefs

Cf. A129404, A129405, A129406, A129407, A129408, A129409, A129410, A129411.

Cf. A129658, A129659, A129660, A129661, A129662, A129664, A129665.

Programs

Mathematica

nmax = 100; prec = 3000 (* Adjust the precision depending on nmax. *); c = N[ 4 Pi^3/(81 Sqrt[3]), prec]; p = First@Transpose@NestList[{Floor[ 1/(1 - #[[1]] #[[2]]) ], 1 - #[[1]] #[[2]]}&, {Floor[1/c], c}, nmax - 1]; p = Drop[ FoldList[Times, 1, p], 1 ]; Denominator[ FoldList[ Plus, 0, (-1)^Range[0, Length[p] - 1]/p ] ]

Formula

chi3(k) = Kronecker(-3, k); chi3(k) is 0, 1, -1 when k reduced modulo 3 is 0, 1, 2, respectively; chi3 is A049347 shifted.

Series: L(3, chi3) = Sum_{k=1..infinity} chi3(k) k^{-3} = 1 - 1/2^3 + 1/4^3 - 1/5^3 + 1/7^3 - 1/8^3 + 1/10^3 - 1/11^3 + ...

Closed form: L(3, chi3) = 4 Pi^3/(81 sqrt(3)).