A158241 -id:A158241 - cp's OEIS frontend

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

Benoit Cloitre, Mar 31 2009

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).

			0.2840790438404122960282...

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

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.

Cf. A001622, A158241, A188593, A019845.

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

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

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

cp's OEIS Frontend

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

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