Gauss–Kuzmin–Wirsing operator

In mathematics, the Gauss–Kuzmin–Wirsing operator, named after Carl Gauss, Rodion Osievich Kuzmin and Eduard Wirsing, occurs in the study of continued fractions; it is also related to the Riemann zeta function.

Introduction

The Gauss–Kuzmin–Wirsing operator 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 first eigenfunction of this operator is

${\frac 1{\ln 2}}\ {\frac 1{1+x}}$

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

$x=[0;a_{1},a_{2},a_{3},\dots ]\,$

is the continued fraction representation of a number 0 < x < 1, then

$h(x)=[0;a_{2},a_{3},\dots ].\,$

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.

Eigenvalues

Let us arrange the eigenvalues of the Gauss–Kuzmin–Wirsing operator according to an absolute value:

$1=|\lambda _{{1}}|\geq |\lambda _{{2}}|\geq |\lambda _{{3}}|\geq \cdots .$

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

$\lim \limits _{{n\rightarrow \infty }}{\frac {\lambda _{{n}}}{\lambda _{{n+1}}}}=-\phi ^{{2}},{\text{ where }}\phi ={\frac {1+{\sqrt {5}}}{2}}.$

In 2014, Giedrius Alkauskas proved this conjecture.^[1] Moreover, the following asymptotic result holds:

$(-1)^{{n+1}}\lambda _{{n}}=\phi ^{{-2n}}+C\cdot {\frac {\phi ^{{-2n}}}{{\sqrt {n}}}}+d(n)\cdot {\frac {\phi ^{{-2n}}}{n}},{\text{ where }}C={\frac {{\sqrt[ {4}]{5}}\cdot \zeta (3/2)}{2{\sqrt {\pi }}}}=1.1019785625880999_{{+}};$

here the function $d(n)$ is bounded, and $\zeta (\star )$ 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 ${\displaystyle \mathbb {N} ^{\omega }}$, the GKW operator can also be viewed as an operator on the function space ${\displaystyle \mathbb {N} ^{\omega }\to \mathbb {C} }$ (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 ${\displaystyle |\lambda |<1}$ of the complex plane. That is, given the cylinder ${\displaystyle C_{n}[b]=\{(a_{1},a_{2},\cdots )\in \mathbb {N} ^{\omega }:a_{n}=b\}}$, the operator G shifts it to the left: ${\displaystyle GC_{n}[b]=C_{n-1}[b]}$. Taking ${\displaystyle r_{n,b}(x)}$ to be the indicator function which is 1 on the cylinder (when ${\displaystyle x\in C_{n}[b]}$), and zero otherwise, one has that ${\displaystyle Gr_{n,b}=r_{n-1,b}}$. The series

${\displaystyle f(x)=\sum _{n=1}^{\infty }\lambda ^{n-1}r_{n,b}(x)}$

then is an eigenfunction with eigenvalue $\lambda$. That is, one has ${\displaystyle [Gf](x)=\lambda f(x)}$ whenever the summation converges: that is, when ${\displaystyle |\lambda |<1}$.

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 ${\displaystyle ?^{\prime }}$. That is, one has that ${\displaystyle G?^{\prime }=?^{\prime }}$.^[2]

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

$\zeta (s)={\frac {s}{s-1}}-s\int _{0}^{1}x\left[Gx^{{s-1}}\right]\,dx$

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

$G_{{mn}}=\sum _{{k=0}}^{n}(-1)^{k}{n \choose k}{k+m+1 \choose m}\left[\zeta (k+m+2)-1\right].$

This operator is extremely well formed, and thus very numerically tractable. Note that each entry is a finite rational zeta series. 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

$\zeta (s)={\frac {s}{s-1}}-s\sum _{{n=0}}^{\infty }(-1)^{n}{s-1 \choose n}t_{n}$

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:

$t_{n}=1-\gamma +\sum _{{k=1}}^{n}(-1)^{k}{n \choose k}\left[{\frac {1}{k}}-{\frac {\zeta (k+1)}{k+1}}\right]$

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.

This expansion of the Riemann zeta is investigated in ^{[3][4][5][6][7]} The coefficients are decreasing as

$\left({\frac {2n}{\pi }}\right)^{{1/4}}e^{{-{\sqrt {4\pi n}}}}\cos \left({\sqrt {4\pi n}}-{\frac {5\pi }{8}}\right)+{\mathcal {O}}\left({\frac {e^{{-{\sqrt {4\pi n}}}}}{n^{{1/4}}}}\right).$

References

1. Giedrius Alkauskas, http://arxiv.org/abs/1210.4083 Transfer operator for the Gauss' continued fraction map. I. Structure of the eigenvalues and trace formulas (2014).
2. Linas Vepstas, "On the Minkowski Measure", (2008) arXiv:0810.1265
3. A. Yu. Eremin, I. E. Kaporin, and M. K. Kerimov, "The calculation of the Riemann zeta-function in the complex domain", U.S.S.R. Comput. Math. and Math. Phys. 25 (1985), no. 2, 111–119
4. A. Yu. Yeremin, I. E. Kaporin, and M. K. Kerimov, "Computation of the derivatives of the Riemann zeta-function in the complex domain", U.S.S.R. Comput. Math. and Math. Phys. 28 (1988), no. 4, 115–124
5. Luis Báez-Duarte, "A New Necessary and Sufficient Condition for the Riemann Hypothesis" (2003) ArXiv math.NT/0307215
6. Luis Báez-Duarte, "A sequential Riesz-like criterion for the Riemann hypothesis", International Journal of Mathematics and Mathematical Sciences, 21, pp. 3527–3537 (2005)
7. Philippe Flajolet and Linas Vepstas, "On differences of zeta values", J. Comput. Appl. Math. 220, No. 1-2, 58-73 (2008).

General references

• A. Ya. Khinchin, Continued Fractions, 1935, English translation University of Chicago Press, 1961 ISBN 0-486-69630-8 (See section 15).
• K. I. Babenko, On a Problem of Gauss, Soviet Mathematical Doklady 19:136–140 (1978) MR 57 #12436
• K. I. Babenko and S. P. Jur'ev, On the Discretization of a Problem of Gauss, Soviet Mathematical Doklady 19:731–735 (1978). MR 81h:65015
• A. Durner, On a Theorem of Gauss–Kuzmin–Lévy. Arch. Math. 58, 251–256, (1992). MR 93c:11056
• A. J. MacLeod, High-Accuracy Numerical Values of the Gauss–Kuzmin Continued Fraction Problem. Computers Math. Appl. 26, 37–44, (1993).
• E. Wirsing, On the Theorem of Gauss–Kuzmin–Lévy and a Frobenius-Type Theorem for Function Spaces. Acta Arith. 24, 507–528, (1974). MR 49 #2637

Further reading

• Keith Briggs, A precise computation of the Gauss–Kuzmin–Wirsing constant (2003) (Contains a very extensive collection of references.)
• Phillipe Flajolet and Brigitte Vallée, On the Gauss–Kuzmin–Wirsing Constant (1995).
• Linas Vepstas The Bernoulli Operator, the Gauss–Kuzmin–Wirsing Operator, and the Riemann Zeta (2004) (PDF)

External links

• Weisstein, Eric W. "Gauss-Kuzmin-Wirsing Constant". MathWorld.

• "Sloane's A038517". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.