Gauss-Kuzmin-Wirsing constant

In mathematics, the Gauss–Kuzmin–Wirsing operator, named after Carl Gauss, Rodion Osievich Kuzmin and Eduard Wirsing, occurs in the study of continued fractions; it is also related to the Riemann zeta function.

The Gauss–Kuzmin–Wirsing operator is the transfer operator of the Gauss map

This operator acts on functions as

The first eigenfunction of this operator is

which corresponds to an eigenvalue of λ₁=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 fractions: if

is the continued fraction representation of a number 0 < x < 1, then

Additional eigenvalues can be computed numerically; the next eigenvalue is λ₂ = −0.3036630029... (sequence in the OEIS) 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:

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

In 2014, Giedrius Alkauskas proved this conjecture. Moreover, the following asymptotic result holds:

here the function ${\displaystyle d(n)}$ is bounded, and ${\displaystyle \zeta (\star )}$ is the Riemann zeta function.

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