Editing Gauss–Kuzmin–Wirsing operator (section)

===Eigenvalues of the operator===

The first [[eigenfunction]] of this operator is

:<math>\frac 1{\ln 2}\ \frac 1{1+x}</math>

which corresponds to an [[eigenvalue]] of ''λ''<sub>1</sub>&nbsp;=&nbsp;1. This eigenfunction gives the probability of the occurrence of a given integer in a continued fraction expansion, and is known as the [[Gauss–Kuzmin distribution]].  This follows in part because the Gauss map acts as a truncating [[shift operator]] for the [[continued fraction]]s: if

: <math>x=[0;a_1,a_2,a_3,\dots]</math>

is the continued fraction representation of a number 0&nbsp;<&nbsp;''x''&nbsp;<&nbsp;1, then

: <math>h(x)=[0;a_2,a_3,\dots].</math>

Because <math>h</math> is conjugate to a [[Bernoulli shift]], the eigenvalue <math>\lambda_1=1</math> is simple, and since the operator leaves invariant the Gauss–Kuzmin measure, the operator is [[ergodic]] with respect to the measure. This fact allows a short proof of the existence of [[Khinchin's constant]].

Additional eigenvalues can be computed numerically; the next eigenvalue is ''λ''<sub>2</sub> = −0.3036630029... {{OEIS|A038517}}
and its absolute value is known as the '''Gauss–Kuzmin–Wirsing constant'''. Analytic forms for additional eigenfunctions are not known.  It is not known if the eigenvalues are [[irrational]].

Let us arrange the eigenvalues of the Gauss–Kuzmin–Wirsing operator according to an absolute value:

:<math>1=|\lambda_1|> |\lambda_2|\geq|\lambda_3|\geq\cdots.</math>

It was conjectured in 1995 by [[Philippe Flajolet]] and [[Brigitte Vallée]] that

:<math>
\lim_{n\to\infty} \frac{\lambda_n}{\lambda_{n+1}} = -\varphi^2, \text{ where } \varphi=\frac{1+\sqrt 5} 2.
</math>

In 2018, Giedrius Alkauskas gave a convincing argument that this conjecture can be refined to a much stronger statement:<ref>{{cite arXiv |last1=Alkauskas |first1=Giedrius |year=2018 |title=Transfer operator for the Gauss' continued fraction map. I. Structure of the eigenvalues and trace formulas |eprint=1210.4083 |class=math.NT}}</ref>

:<math>
\begin{align}
& (-1)^{n+1}\lambda_n=\varphi^{-2n} + C\cdot\frac{\varphi^{-2n}}{\sqrt{n}}+d(n)\cdot\frac{\varphi^{-2n}}{n}, \\[4pt]
& \text{where } C=\frac{\sqrt[4]{5}\cdot\zeta(3/2)}{2\sqrt{\pi}}=1.1019785625880999_{+};
\end{align}
</math>

here the function <math>d(n)</math> is bounded, and <math>\zeta(\star)</math> is the [[Riemann zeta function]].