Gauss–Kuzmin–Wirsing operator

"GKW" redirects here. For the Indian engineering firm see Guest Keen Williams.

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

${\displaystyle h(x)=1/x-\lfloor 1/x\rfloor .\,}$

This operator acts on functions as

${\displaystyle [Gf](x)=\sum _{n=1}^{\infty }{\frac {1}{(x+n)^{2}}}f\left({\frac {1}{x+n}}\right).}$

The first eigenfunction of this operator is

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

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

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

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

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

Additional eigenvalues can be computed numerically; the next eigenvalue is λ2 = −0.3036630029... (sequence A038517 in 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:

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

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

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

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

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

here the function ${\displaystyle d(n)}$ is bounded, and ${\displaystyle \zeta (\star )}$ is the Riemann zeta function.

Relationship to the Riemann zeta

The GKW operator is related to the Riemann zeta function. Note that the zeta can be written as

${\displaystyle \zeta (s)={\frac {1}{s-1}}-s\int _{0}^{1}h(x)x^{s-1}\;dx}$

which implies that

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

by change-of-variable.

Matrix elements

Consider the Taylor series expansions at x=1 for a function f(x) and ${\displaystyle g(x)=[Gf](x)}$. That is, let

${\displaystyle f(1-x)=\sum _{n=0}^{\infty }(-x)^{n}{\frac {f^{(n)}(1)}{n!}}}$

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

${\displaystyle (-1)^{m}{\frac {g^{(m)}(1)}{m!}}=\sum _{n=0}^{\infty }G_{mn}(-1)^{n}{\frac {f^{(n)}(1)}{n!}},}$

where the matrix elements of the GKW operator are given by

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

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

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

where the ${\displaystyle t_{n}}$ are given by the matrix elements above:

${\displaystyle t_{n}=\sum _{m=0}^{\infty }{\frac {G_{mn}}{(m+1)(m+2)}}.}$

Performing the summations, one gets:

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

where ${\displaystyle \gamma }$ is the Euler–Mascheroni constant. These ${\displaystyle t_{n}}$ play the analog of the Stieltjes constants, but for the falling factorial expansion. By writing

${\displaystyle a_{n}=t_{n}-{\frac {1}{2(n+1)}}}$

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 [2][3][4][5][6] The coefficients are decreasing as

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

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. 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
3. 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
4. Luis Báez-Duarte, "A New Necessary and Sufficient Condition for the Riemann Hypothesis" (2003) ArXiv math.NT/0307215
5. Luis Báez-Duarte, "A sequential Riesz-like criterion for the Riemann hypothesis", Internation Journal of Mathematics and Mathematical Sciences, 21, pp. 3527–3537 (2005)
6. 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