In mathematics, the Gauss–Kuzmin–Wirsing operator is the transfer operator of the Gauss map. 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 fractionsEdit
The Gauss mapEdit
The Gauss function (map) h is :
- denotes floor function
Operator on the mapsEdit
The Gauss–Kuzmin–Wirsing operator acts on functions as
Eigenvalues of the operatorEdit
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 A038517 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:
In 2014, Giedrius Alkauskas proved this conjecture. Moreover, the following asymptotic result holds:
here the function is bounded, and is the Riemann zeta function.
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 to be the indicator functions on the cylinders of the product topology). 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 (when ), 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 .
Relationship to the Riemann zeta functionEdit
The GKW operator is related to the Riemann zeta function. Note that the zeta function can be written as
which implies that
Consider the Taylor series expansions at x=1 for a function f(x) and . That is, let
and write likewise for g(x). 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.
The Riemann zeta can be written as
where the are given by the matrix elements above:
Performing the summations, one gets:
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.
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