Lehmer,s Constant
المؤلف:
Le Lionnais, F
المصدر:
Les nombres remarquables. Paris: Hermann
الجزء والصفحة:
...
2-3-2020
955
Lehmer's Constant
The Lehmer cotangent expansion for which the convergence is slowest occurs when the inequality in the recurrence equation
 |
(1)
|
for
![x=cot[sum_(k=0)^infty(-1)^kcot^(-1)b_k]](http://mathworld.wolfram.com/images/equations/LehmersConstant/NumberedEquation2.gif) |
(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
(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."
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