A129662 Numerators 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.

0, 1, 7, 23, 1471, 94145, 327200947, 6435419387591, 3576528877557150803, 528385432191928134753762821, 98874483030041554423376610821029, 1056201236231124272980670932252118118619723

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 ]; Numerator[ 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, A129663, A129664, A129665.

Sequence in context: A228699 A159485 A009047 * A012482 A124985 A349078

Adjacent sequences: A129659 A129660 A129661 * A129663 A129664 A129665

Keyword

nonn,frac,easy

Author

Stuart Clary, Apr 30 2007

Status

approved