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 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 2014, Giedrius Alkauskas proved this conjecture. Moreover, the following asymptotic result holds: 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 . In the later case, it has a continuous spectrum, with eigenvalues in the unit disk of the complex plane. That is, given the cylinder, the operator G shifts it to the left:. Taking to be the indicator function which is 1 on the cylinder, and zero otherwise, one has that. The series then is an eigenfunction with eigenvalue. That is, one has whenever the summation converges: that is, when. A special case arises when one wishes to consider the Haar measure of the shift operator, that is, a function that is invariant under shifts. This is given by the Minkowski measure. That is, one has that.
The GKW operator is related to the Riemann zeta function. Note that the zeta function can be written as which implies that by change-of-variable.
Matrix elements
Consider the Taylor series expansions at x=1 for a function f and. That is, let and write likewise for g. The expansion is made about x = 1 because the GKW operator is poorly behaved at x = 0. The expansion is made about 1-x so that we can keep x a positive number, 0 ≤ x ≤ 1. Then the GKW operator acts on the Taylor coefficients as where the matrix elements of the GKW operator are given by This operator is extremely well formed, and thus very numerically tractable. The Gauss–Kuzmin constant is easily computed to high precision by numerically diagonalizing the upper-left n by n portion. There is no known closed-form expression that diagonalizes this operator; that is, there are no closed-form expressions known for the eigenvectors.
Riemann zeta
The Riemann zeta can be written as where the are given by the matrix elements above: Performing the summations, one gets: where is the Euler–Mascheroni constant. These play the analog of the Stieltjes constants, but for the falling factorial expansion. By writing one gets: a0 = −0.0772156... and a1 = −0.00474863... and so on. The values get small quickly but are oscillatory. Some explicit sums on these values can be performed. They can be explicitly related to the Stieltjes constants by re-expressing the falling factorial as a polynomial with Stirling number coefficients, and then solving. More generally, the Riemann zeta can be re-expressed as an expansion in terms of Sheffer sequences of polynomials. This expansion of the Riemann zeta is investigated in the following references. The coefficients are decreasing as
