Gauss-kuzmin-wirsing Operator

In mathematics, the Gauss-Kuzmin-Wirsing operator occurs in the study of continued fractions; it is also related to the Riemann zeta function. It is the transfer operator of the Gauss map
h(x)=1/x-\lfloor 1/x \rfloor
This operator acts on functions as
Gf(x) = \sum_{n=1}^\infty \frac {1}{(x+n)^2} f \left(\frac {1}{x+n}\right)
The zeroth eigenfunction of this operator is
\frac {\ln 2} {1+x}
which corresponds to an eigenvalue of 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 x=0;a_1,a_2,a_3,... is the continued fraction representation of a number 0 < x < 1, then h(x)=0;a_2,a_3,.... Additional eigenvalues can be computed numerically; the next eigenvalue is \lambda_1=0.3036630029... and 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.

Relationship to the Riemann zeta

The GKW operator is related to the Riemann zeta function. Note that the zeta can be written as
\zeta(s)=\frac{1}{s-1}-s\int_0^1 h(x) x^{s-1} \; dx
which implies that
\zeta(s)=\frac{s}{s-1}-s\int_0^1 dx\; x \left\right
by change-of-variable.

Matrix Elements

Consider the Taylor series expansions at x=1 for a function f(x) and g(x)=Gf(x). That is, let
f(1-x)=\sum_{n=0}^\infty (-x)^n \frac{f^{(n)}(1)}{n!}
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
(-1)^m \frac{g^{(m)}(1)}{m!} = \sum_{n=0}^\infty G_{mn} (-1)^n \frac{f^{(n)}(1)}{n!}
where the matrix elements of the GKW operator are given by
G_{mn}=\sum_{k=0}^n (-1)^k {n \choose k} {k+m+1 \choose m} \zeta (k+m+2)
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 eigenvalues or eigenvectors.

Riemann zeta

The Riemann zeta can be written as
\zeta(s)=\frac{s}{s-1}-s \sum_{n=0}^\infty (-)^n {s-1 \choose n} t_n
where the t_n are given by the matrix elements above:
t_n=\sum_{m=0}^\infty \frac{G_{mn}} {(m+1)(m+2)}
Performing the summations, one gets:
t_n=1-\gamma + \sum_{k=1}^n (-1)^n {n \choose k} \left+ \frac {\zeta(k+1)} {k+1} \right
where \gamma is the Euler-Mascheroni constant. These t_n play the analog of the Stieltjes constants, but for the falling factorial expansion. By writing
a_n=t_n - \frac{1}{2(n+1)}
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.

References

 

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