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, ... (Sloane's A002065), and defines the constant known as Lehmer's constant as

			(4)
			(5)
			(6)

(Sloane's A030125).

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

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."

CITE THIS AS:

Weisstein, Eric W. "Lehmer's Constant." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/LehmersConstant.html