%I #29 Nov 20 2024 23:48:51

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

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

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

%N Decimal expansion of theta = arctan((sqrt(10-2*sqrt(5))-2)/(sqrt(5)-1)).

%C This number arose in the Davenport-Heilbronn zeta-function which satisfies a functional equation (like zeta) but does not satisfy RH. Some nontrivial zeros are off the critical line (see reference).

%D P. Borwein et al., The Riemann Hypothesis, Springer (2009), 136-137.

%H E. Bombieri and D. Hejhal, <a href="http://gallica.bnf.fr/ark:/12148/bpt6k5744593d/f13.image">Sur les zéros des fonctions zeta d'Epstein</a>, (mostly in English) Comptes rendus de l'Académie des Sciences, Paris, 304 (1987), 213-217.

%H H. Davenport and H. Heilbronn, <a href="https://doi.org/10.1112/jlms/s1-11.3.181">On the zeros of certain Dirichlet series I</a>, J. London Math. Soc. 11 (1936), 181-185.

%H H. Davenport and H. Heilbronn, <a href="https://doi.org/10.1112/jlms/s1-11.4.307">On the zeros of certain Dirichlet series II</a>, J. London Math. Soc. 11 (1936), 307-312.

%H <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>

%e 0.27678717...

%t RealDigits[ArcTan[(Sqrt[10-2*Sqrt[5]]-2)/(Sqrt[5]-1)],10,120][[1]] (* _Harvey P. Dale_, Mar 03 2018 *)

%o (PARI) atan((sqrt(10-2*sqrt(5))-2)/(sqrt(5)-1)) \\ _Charles R Greathouse IV_, Mar 10 2016

%K cons,nonn

%O 0,1

%A _Benoit Cloitre_, Mar 14 2009

%E Keyword:cons inserted, leading zero and offset adjusted by _R. J. Mathar_, Jul 15 2010