Lehmer's constant is the value of the Lehmer cotangent expansion with the slowest convergence. It occurs when the inequality
in the recurrence equation
(1)
for
(2)
is replaced by equality, giving and
(3)
for .
This recurrence gives values of corresponding to 0, 1, 3, 13, 183, 33673, ... (OEIS A002065),
and defines the constant known as Lehmer's constant as
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, France: 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." Monatsh.
Math.150, 49-71, 2007. https://doi.org/10.1007/s00605-006-0415-7.Sloane,
N. J. A. Sequences A002065/M2961
and A030125 in "The On-Line Encyclopedia
of Integer Sequences."
![x=cot[sum_(k=0)^infty(-1)^kcot^(-1)b_k]](/images/equations/LehmersConstant/NumberedEquation2.svg)
and
.
corresponding to 0, 1, 3, 13, 183, 33673, ... (OEIS A002065),
and defines the constant known as Lehmer's constant as
is a transcendental
number (Rivoal 2007, Theorem 8). Lehmer (1938) proved that it is not an
algebraic number of degree
less than 4.
