A153072 Continued fraction for L(3, chi4), where L(s, chi4) is the Dirichlet L-function for the non-principal character modulo 4

0, 1, 31, 4, 1, 18, 21, 1, 1, 2, 1, 2, 1, 3, 6, 3, 28, 1, 3, 2, 1, 2, 21, 1, 1, 32, 1, 1, 1, 5, 3, 1, 2, 1, 27, 11, 1, 2, 1, 5, 1, 3, 4, 3, 1, 4, 1, 1, 2, 1, 9, 8, 1, 2, 2, 1, 14, 2, 1, 7, 2, 2, 1, 20, 2, 1, 5, 10, 1, 4, 2, 2, 1, 2, 106, 4, 1, 1, 1, 1, 1, 10, 9, 3, 3, 14

Offset

0,3

References

Leonhard Euler, "Introductio in Analysin Infinitorum", First Part, Articles 175, 284 and 287.

Bruce C. Berndt, "Ramanujan's Notebooks, Part II", Springer-Verlag, 1989. See page 293, Entry 25 (iii).

Links

Table of n, a(n) for n=0..85.

Formula

chi4(k) = Kronecker(-4, k); chi4(k) is 0, 1, 0, -1 when k reduced modulo 4 is 0, 1, 2, 3, respectively; chi4 is A101455.

Series: L(3, chi4) = Sum_{k>=1} chi4(k) k^{-3} = 1 - 1/3^3 + 1/5^3 - 1/7^3 + 1/9^3 - 1/11^3 + 1/13^3 - 1/15^3 + ...

Series: L(3, chi4) = Sum_{k>=0} tanh((2k+1) Pi/2)/(2k+1)^3. [Ramanujan; see Berndt, page 293]

Closed form: L(3, chi4) = Pi^3/32.

Example

L(3, chi4) = 0.9689461462593693804836348458469186... = [0; 1, 31, 4, 1, 18, 21, 1, 1, 2, 1, 2, 1, 3, 6, 3, 28, ...].

Mathematica

nmax = 1000; ContinuedFraction[Pi^3/32, nmax + 1]

Crossrefs

Cf. A153071, A153073, A153074.

Sequence in context: A128372 A334061 A040942 * A040943 A040944 A040940

Adjacent sequences: A153069 A153070 A153071 * A153073 A153074 A153075

Keyword

nonn,cofr,easy

Author

Stuart Clary, Dec 17 2008

Status

approved