# Gauss–Kuzmin–Wirsing operator

# Gauss–Kuzmin–Wirsing operator

In mathematics, the **Gauss–Kuzmin–Wirsing operator** is the transfer operator of the Gauss map.

Name

It is named after:

Carl Gauss

Rodion Osievich Kuzmin

Eduard Wirsing

Importance

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 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.^{[1]}

Operator on the maps

`The Gauss–Kuzmin–Wirsingoperatoracts on functionsas`

Eigenvalues of the operator

The first eigenfunction of this operator is

which corresponds to an eigenvalue of *λ*1=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 *λ*2 = −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:

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

In 2014, Giedrius Alkauskas proved this conjecture.^{[2]} Moreover, the following asymptotic result holds:

`here the functionis bounded, andis theRiemann 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 onBaire space, the GKW operator can also be viewed as an operator on the function space(considered as aBanach space, with basis functions taken to be theindicator functionson thecylindersof theproduct topology). In the later case, it has a continuous spectrum, with eigenvalues in the unit diskof the complex plane. That is, given the cylinder, the operator G shifts it to the left:. Takingto 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 haswhenever the summation converges: that is, when.`

`A special case arises when one wishes to consider theHaar measureof the shift operator, that is, a function that is invariant under shifts. This is given by theMinkowski measure. That is, one has that.`

^{[3]}Relationship to the Riemann zeta

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 theTaylor seriesexpansions at x=1 for a function`

*f*(*x*) and. That is, letand 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.

Riemann zeta

The Riemann zeta can be written as

`where theare given by the matrix elements above:`

Performing the summations, one gets:

`whereis theEuler–Mascheroni constant. Theseplay the analog of theStieltjes constants, but for thefalling factorialexpansion. By writing`

one gets: *a*0 = −0.0772156... and *a*1 = −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.^{[4]}^{[5]}^{[6]}^{[7]}^{[8]} The coefficients are decreasing as

## References

*USSR Comput. Math. and Math. Phys*.

**25**(2): 111–119.

*USSR Comput. Math. and Math. Phys*.

**28**(4): 115–124.

*International Journal of Mathematics and Mathematical Sciences*.

**21**: 3527–3537.

*Journal of Computational and Applied Mathematics*.

**220**: 58. arXiv:math.CA/0611332. Bibcode:2008JCoAM.220...58F. doi:10.1016/j.cam.2007.07.040.