A158241 Decimal expansion of theta = arctan((sqrt(10-2*sqrt(5))-2)/(sqrt(5)-1)).

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, 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, 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

Offset: 0

Author

Benoit Cloitre, Mar 14 2009

Keywords

• cons
• nonn

Comments

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).

Examples

			0.27678717...
		

References

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

Links

• E. Bombieri and D. Hejhal, Sur les zéros des fonctions zeta d'Epstein, (mostly in English) Comptes rendus de l'Académie des Sciences, Paris, 304 (1987), 213-217.
• H. Davenport and H. Heilbronn, On the zeros of certain Dirichlet series I, J. London Math. Soc. 11 (1936), 181-185.
• H. Davenport and H. Heilbronn, On the zeros of certain Dirichlet series II, J. London Math. Soc. 11 (1936), 307-312.
• Index entries for transcendental numbers

Programs

Mathematica

RealDigits[ArcTan[(Sqrt[10-2*Sqrt[5]]-2)/(Sqrt[5]-1)],10,120][[1]] (* Harvey P. Dale, Mar 03 2018 *)

PARI

atan((sqrt(10-2*sqrt(5))-2)/(sqrt(5)-1)) \\ Charles R Greathouse IV, Mar 10 2016

Extensions

Keyword:cons inserted, leading zero and offset adjusted by R. J. Mathar, Jul 15 2010