# Definition:Lehmer's Constant/Mistake

## Source Work

Thème et variations
$0,59263 27182 \ldots$

## Mistake

• Constante de Lehmer (1938)
$= \map {\operatorname {cotg} } {\operatorname {Arcotg} 1 - \operatorname {Arcotg} 3 + \operatorname {Arcotg} {13} - \operatorname {Arcotg} {183} + \operatorname {Arcotg} {33973} - \dotsb + \paren {-1}^n \operatorname {Arcotg} u_n - \dotsb}$. La suite $u_n$ est définie par $u_0 = 1; u_{n + 1} = u_n^2 + u_n + 1$.

That is, in English:

• Lehmer's constant (1938)
$= \map \cot {\arccot 1 - \arccot 3 + \arccot {13} - \arccot {183} + \arccot {33973} - \dotsb + \paren {-1}^n \arccot u_n - \dotsb}$. The sequence $u_n$ is defined by $u_0 = 1; u_{n + 1} = u_n^2 + u_n + 1$.

## Correction

$= \map {\operatorname {cotg} } {\operatorname {Arcotg} 0 - \operatorname {Arcotg} 1 + \operatorname {Arcotg} 3 - \operatorname {Arcotg} {13} + \operatorname {Arcotg} {183} - \operatorname {Arcotg} {33973} + \dotsb + \paren {-1}^n \operatorname {Arcotg} u_n - \dotsb}$. La suite $u_n$ est définie par $u_0 = 1; u_{n + 1} = u_n^2 + u_n + 1$.

that is:

$= \map \cot {\arccot 0 - \arccot 1 + \arccot 3 - \arccot {13} + \arccot {183} - \arccot {33973} + \dotsb + \paren {-1}^n \arccot u_n - \dotsb}$. The sequence $u_n$ is defined by $u_0 = 0; u_{n + 1} = u_n^2 + u_n + 1$.

It is to be noted that $\arccot 0$ frequently throws an error on an electronic calculator because $\arccot x$ is calculated on such machines as $\arctan \dfrac 1 x$ which of course is not defined for $x = 0$.

But $\arccot 0$ is in fact $\dfrac \pi 2$, and so should really be calculated.