A158934 - OEIS

%I #21 Jan 23 2022 07:28:06

%S 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,

%T 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,

%U 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

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

%C 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).

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

%H Bruce C. Berndt, Heng Huat Chan and Liang-Cheng Zhang, <a href="https://doi.org/10.1515/crll.1996.480.141">Explicit evaluations of the Rogers-Ramanujan continued fraction</a>, Journal für die reine und angewandte Mathematik, Vol. 480 (1996), pp. 141-160, eq. (1.1).

%H Harold Davenport and Hans Heilbronn, <a href="https://doi.org/10.1112/jlms/s1-11.3.181">On the zeros of certain Dirichlet series</a>, Journal of the London Mathematical Society, Vol. s1-11, No. 3 (1936), pp. 181-185.

%H Harold Davenport and Hans Heilbronn, <a href="https://doi.org/10.1112/jlms/s1-11.4.307">On the zeros of certain Dirichlet series (Second paper)</a>, Journal of the London Mathematical Society, Vol. s1-11, No. 4 (1936), pp. 307-312.

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

%F Equals (A001622-1)/(2*A019845+1). - _R. J. Mathar_, Apr 02 2009

%F Equals sqrt((5 + sqrt(5))/2) - (sqrt(5) + 1)/2 = A188593 - A001622. - _Amiram Eldar_, Jan 23 2022

%e 0.2840790438404122960282...

%t (Sqrt[5]-1) / (2+Sqrt[10-2*Sqrt[5]]) // RealDigits[#, 10, 104]& // First (* _Jean-François Alcover_, Mar 04 2013 *)

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

%Y Cf. A001622, A158241, A188593, A019845.

%K cons,nonn

%O 0,1

%A _Benoit Cloitre_, Mar 31 2009