Lehmer's Constant

N.B. A detailed on-line essay by S. Finch was the starting point for this entry.

Lehmer (1938) showed that every Positive Irrational Number $x$ has a unique infinite continued cotangent representation of the form

$$x=\cot\left[{\,\sum_{k=0}^\infty (-1)^k\cot^{-1} b_k}\right],$$

where the $b_k$s are Nonnegative and

$$b_k\geq (b_{k-1})^2+b_{k-1}+1.$$

The case for which the convergence is slowest occurs when the inequality is replaced by equality, giving $c_0=0$ and

$$c_k=(c_{k-1})^2+c_{k-1}+1$$

for $k\geq 1$. The first few values are $c_k$ are 0, 1, 3, 13, 183, 33673, ... (Sloane's A024556), resulting in the constant

$$\xi = \cot(\cot^{-1} 0-\cot^{-1}1+\cot^{-1}3-\cot^{-1}13$$

$$\phantom{=} +\cot^{-1}183-\cot^{-1}33673+\cot^{-1}1133904603$$

$$\phantom{=} -\cot^{-1}1285739649838492213+\ldots+(-1)^kc_k+\ldots)$$

$$= \cot({\textstyle{1\over 4}}\pi+\cot^{-1}3-\cot^{-1}13$$

$$\phantom{=} +\cot^{-1}183-\cot^{-1}33673+\cot^{-1}1133904603$$

$$\phantom{=} -\cot^{-1}1285739649838492213+\ldots+(-1)^kc_k+\ldots)$$

$$= 0.59263271\ldots$$

(Sloane's A030125). $\xi$ is not an Algebraic Number of degree less than $4$, but Lehmer's approach cannot show whether or not $\xi$ is Transcendental.

See also Algebraic Number, Transcendental Number

References

Finch, S. ``Favorite Mathematical Constants.'' http://www.mathsoft.com/asolve/constant/lehmer/lehmer.html

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.

Plouffe, S. ``The Lehmer Constant.'' http://www.lacim.uqam.ca/piDATA/lehmer.txt.

Sloane, N. J. A. A024556 and A030125 in ``An On-Line Version of the Encyclopedia of Integer Sequences.'' http://www.research.att.com/~njas/sequences/eisonline.html.