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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. 15
0, 1, 7, 23, 1471, 94145, 327200947, 6435419387591, 3576528877557150803, 528385432191928134753762821, 98874483030041554423376610821029, 1056201236231124272980670932252118118619723 (list; graph; refs; listen; history; text; internal format)
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 A126612

Adjacent sequences:  A129659 A129660 A129661 * A129663 A129664 A129665

KEYWORD

nonn,frac,easy

AUTHOR

Stuart Clary, Apr 30 2007

STATUS

approved

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Last modified June 18 13:35 EDT 2021. Contains 345112 sequences. (Running on oeis4.)