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

0,3

References

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

Links

Table of n, a(n) for n=0..11.

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)).

Example

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, ...

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 ] ]

Crossrefs

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

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

Sequence in context: A203635 A000810 A171740 * A112646 A119522 A070494

Adjacent sequences: A129660 A129661 A129662 * A129664 A129665 A129666

Keyword

nonn,frac,easy

Author

Stuart Clary, Apr 30 2007

Status

approved