Lehmer's Constant

The Lehmer cotangent expansion for which the convergence is slowest occurs when the inequality in the recurrence equation

(1)

for

(2)

is replaced by equality, giving  and

(3)

for .

This recurrences gives values of  corresponding to 0, 1, 3, 13, 183, 33673, ... (OEIS A002065), and defines the constant known as Lehmer's constant as

			(4)
			(5)
			(6)

(OEIS A030125).

 is not an algebraic number of degree less than 4, but Lehmer's approach cannot show whether  is transcendental.

REFERENCES:

Finch, S. R. "Lehmer's Constant." §6.6. in Mathematical Constants. Cambridge, England: Cambridge University Press, pp. 433-434, 2003.

Le Lionnais, F. Les nombres remarquables. Paris: Hermann, p. 29, 1983.

Lehmer, D. H. "A Cotangent Analogue of Continued Fractions." Duke Math. J. 4, 323-340, 1938.

Rivoal, T. "Propriétés diophantiennes du développement en cotangente continue de Lehmer." http://www-fourier.ujf-grenoble.fr/~rivoal/articles/cotan.pdf.

Sloane, N. J. A. Sequences A002065/M2961 and A030125 in "The On-Line Encyclopedia of Integer Sequences."