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
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).
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
KEYWORD
cons,nonn
AUTHOR
Benoit Cloitre, Mar 31 2009
STATUS
approved