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Lehmer's Constant

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

b_k>=b_(k-1)^2+b_(k-1)+1.
(1)

for

x=cot[sum_(k=0)^infty(-1)^kcot^(-1)b_k]
(2)

is replaced by equality, giving c_0=0 and

c_k=c_(k-1)^2+c_(k-1)+1
(3)

for k>=1.

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

xi=cot(cot^(-1)0-cot^(-1)1+cot^(-1)3-cot^(-1)13+cot^(-1)183-cot^(-1)33673+cot^(-1)1133904603-cot^(-1)1285739649838492213+...+(-1)^kc_k+...)
(4)
=cot(1/4pi+cot^(-1)3-cot^(-1)13+cot^(-1)183-cot^(-1)33673+cot^(-1)1133904603-cot^(-1)1285739649838492213+...+(-1)^kc_k+...)
(5)
=0.59263271...
(6)

(OEIS A030125).

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

See also

Algebraic Number, Cotangent, Inverse Cotangent, Lehmer Cotangent Expansion, Transcendental Number

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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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Lehmer's Constant

Cite this as:

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

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