In mathematics, the Gauss–Kuzmin–Wirsing operator is the transfer operator of the Gauss map that takes a positive number to the fractional part of its reciprocal. (This is not the same as the Gauss map in differential geometry.) It is named after Carl Gauss, Rodion Kuzmin, and Eduard Wirsing. It occurs in the study of continued fractions; it is also related to the Riemann zeta function. Relationship to the maps and continued fractions The Gauss map The Gauss function (map) h is : where denotes the floor function. It has an infinite number of jump discontinuities at x = 1/n, for positive integers n. It is hard to approximate it by a single smooth polynomial. Operator on the maps The Gauss–Kuzmin–Wirsing operator acts on functions as Eigenvalues of the operator 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 Because is conjugate to a Bernoulli shift, the eigenvalue 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 λ2 = −0.3036630029... 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 2018, Giedrius Alkauskas gave a convincing argument that this conjecture can be refined to a much stronger statement: here the function is bounded, and is the Riemann zeta function. Continuous spectrum The eigenvalues form a discrete spectrum, when the operator is limited to act on functions on the unit interval of the real number line. More broadly, since the Gauss map is the shift operator on Baire space , the GKW operator can also be viewed as an operator on the function space (considered as a Banach space, with basis functions taken