A158934 Decimal expansion of xi = (cos(Pi/5) - 1/2) / (sin(Pi/5) + 1/2).

2, 8, 4, 0, 7, 9, 0, 4, 3, 8, 4, 0, 4, 1, 2, 2, 9, 6, 0, 2, 8, 2, 9, 1, 8, 3, 2, 3, 9, 3, 1, 2, 6, 1, 6, 9, 0, 9, 1, 0, 8, 8, 0, 8, 8, 4, 4, 5, 7, 3, 7, 5, 8, 2, 7, 5, 9, 1, 6, 2, 6, 6, 6, 1, 5, 5, 0, 4, 5, 8, 7, 7, 3, 5, 1, 4, 8, 4, 5, 5, 3, 7, 3, 0, 3, 7, 8, 4, 1, 7, 7, 5, 2, 2, 3, 1, 6, 2, 5, 8, 6, 7, 0, 4

Offset

0,1

Comments

This constant xi arises in the Davenport-Heilbronn zeta-function Z(s)=Sum_{k>=1} b(k)/k^s where b(k) is the 5-periodic sequence with period [1,xi,-xi,0]. Z satisfies a functional equation (like zeta) but does not satisfy RH. Some nontrivial zeros are off the critical line (see reference).

References

Peter Borwein, Stephen Choi, Brendan Rooney and Andrea Weirathmueller, The Riemann Hypothesis, Springer, 2009, pp. 136-137.

Links

Table of n, a(n) for n=0..103.

Bruce C. Berndt, Heng Huat Chan and Liang-Cheng Zhang, Explicit evaluations of the Rogers-Ramanujan continued fraction, Journal für die reine und angewandte Mathematik, Vol. 480 (1996), pp. 141-160, eq. (1.1).

Harold Davenport and Hans Heilbronn, On the zeros of certain Dirichlet series, Journal of the London Mathematical Society, Vol. s1-11, No. 3 (1936), pp. 181-185.

Harold Davenport and Hans Heilbronn, On the zeros of certain Dirichlet series (Second paper), Journal of the London Mathematical Society, Vol. s1-11, No. 4 (1936), pp. 307-312.

Formula

Equals (sqrt(10-2*sqrt(5))-2)/(sqrt(5)-1).

Equals (A001622-1)/(2*A019845+1). - R. J. Mathar, Apr 02 2009

Equals sqrt((5 + sqrt(5))/2) - (sqrt(5) + 1)/2 = A188593 - A001622. - Amiram Eldar, Jan 23 2022

Example

0.2840790438404122960282...

Mathematica

(Sqrt[5]-1) / (2+Sqrt[10-2*Sqrt[5]]) // RealDigits[#, 10, 104]& // First (* Jean-François Alcover, Mar 04 2013 *)

Prog

(PARI) xi=(cos(Pi/5)-1/2)/(sin(Pi/5)+1/2)

Crossrefs

Cf. A001622, A158241, A188593, A019845.

Sequence in context: A076588 A068565 A092042 * A021356 A030345 A264818

Adjacent sequences: A158931 A158932 A158933 * A158935 A158936 A158937

Keyword

cons,nonn

Author

Benoit Cloitre, Mar 31 2009

Status

approved