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In mathematics, the Gauss–Kuzmin–Wirsing operator is the transfer operator of the Gauss map that takes a positive number to the fractional part of its reciprocal. (This is not the same as the Gauss map in differential geometry.) 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 fractions


The Gauss map

The Gauss function (map) ''h'' is : :h(x)=1/x-\lfloor 1/x \rfloor. where \lfloor 1/x \rfloor denotes the
floor function In mathematics and computer science, the floor function is the function that takes as input a real number , and gives as output the greatest integer less than or equal to , denoted or . Similarly, the ceiling function maps to the least ...
. 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.


Operator on the maps

The Gauss–Kuzmin–Wirsing
operator Operator may refer to: Mathematics * A symbol indicating a mathematical operation * Logical operator or logical connective in mathematical logic * Operator (mathematics), mapping that acts on elements of a space to produce elements of another ...
G acts on functions f as : fx) = \int_0^1 \delta(x-h(y)) f(y) d y = \sum_^\infty \frac f \left(\frac \right).


Eigenvalues of the operator

The first
eigenfunction In mathematics, an eigenfunction of a linear operator ''D'' defined on some function space is any non-zero function f in that space that, when acted upon by ''D'', is only multiplied by some scaling factor called an eigenvalue. As an equation, ...
of this operator is :\frac 1\ \frac 1 which corresponds to an
eigenvalue In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denot ...
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 In mathematics, and in particular functional analysis, the shift operator also known as translation operator is an operator that takes a function to its translation . In time series analysis, the shift operator is called the lag operator. Shift ...
for the
continued fraction In mathematics, a continued fraction is an expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number, then writing this other number as the sum of its integ ...
s: if : x= ;a_1,a_2,a_3,\dots/math> is the continued fraction representation of a number 0 < ''x'' < 1, then : h(x)= ;a_2,a_3,\dots Because h is conjugate to a Bernoulli shift, the eigenvalue \lambda_1=1 is simple, and since the operator leaves invariant the Gauss–Kuzmin measure, the operator is
ergodic In mathematics, ergodicity expresses the idea that a point of a moving system, either a dynamical system or a stochastic process, will eventually visit all parts of the space that the system moves in, in a uniform and random sense. This implies t ...
with respect to the measure. This fact allows a short proof of the existence of
Khinchin's constant In number theory, Aleksandr Yakovlevich Khinchin proved that for almost all real numbers ''x'', coefficients ''a'i'' of the continued fraction expansion of ''x'' have a finite geometric mean that is independent of the value of ''x'' and is kno ...
. Additional eigenvalues can be computed numerically; the next eigenvalue is ''λ''2 = −0.3036630029... 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 Irrationality is cognition, thinking, talking, or acting without inclusion of rationality. It is more specifically described as an action or opinion given through inadequate use of reason, or through emotional distress or cognitive deficiency. ...
. Let us arrange the eigenvalues of the Gauss–Kuzmin–Wirsing operator according to an absolute value: :1=, \lambda_, > , \lambda_, \geq, \lambda_, \geq\cdots. It was conjectured in 1995 by Philippe Flajolet and Brigitte Vallée that :\lim\limits_\frac=-\varphi^2,\text\varphi=\frac. In 2018, Giedrius Alkauskas gave a convincing argument that this conjecture can be refined to a much stronger statement: :(-1)^\lambda_n=\varphi^ + C\cdot\frac+d(n)\cdot\frac, \text C=\frac=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 In mathematics, a topological space X is said to be a Baire space if countable unions of closed sets with empty interior also have empty interior. According to the Baire category theorem, compact Hausdorff spaces and complete metric spaces are ...
\mathbb^\omega, the GKW operator can also be viewed as an operator on the function space \mathbb^\omega\to\mathbb (considered as a
Banach space In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between ve ...
, with basis functions taken to be the
indicator function In mathematics, an indicator function or a characteristic function of a subset of a set is a function that maps elements of the subset to one, and all other elements to zero. That is, if is a subset of some set , one has \mathbf_(x)=1 if x ...
s on the
cylinders A cylinder (from ) has traditionally been a three-dimensional solid, one of the most basic of curvilinear geometric shapes. In elementary geometry, it is considered a prism with a circle as its base. A cylinder may also be defined as an infini ...
of the
product topology In topology and related areas of mathematics, a product space is the Cartesian product of a family of topological spaces equipped with a natural topology called the product topology. This topology differs from another, perhaps more natural-seem ...
). In the later case, it has a continuous spectrum, with eigenvalues in the unit disk , \lambda, <1 of the complex plane. That is, given the cylinder C_n \, the operator G shifts it to the left: GC_n = C_ /math>. Taking r_(x) to be the indicator function which is 1 on the cylinder (when x\in C_n /math>), and zero otherwise, one has that Gr_=r_. The series :f(x)=\sum_^\infty \lambda^ r_(x) then is an eigenfunction with eigenvalue \lambda. That is, one has fx)=\lambda f(x) whenever the summation converges: that is, when , \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 ?^\prime. That is, one has that G?^\prime = ?^\prime.


Relationship to the Riemann zeta function

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


Matrix elements

Consider the
Taylor series In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor se ...
expansions at ''x'' = 1 for a function ''f''(''x'') and g(x)= fx). That is, let :f(1-x)=\sum_^\infty (-x)^n \frac 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 = \sum_^\infty G_ (-1)^n \frac, where the matrix elements of the GKW operator are given by :G_=\sum_^n (-1)^k \left \zeta (k+m+2)- 1\right 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 :\zeta(s)=\frac-s \sum_^\infty (-1)^n t_n where the t_n are given by the matrix elements above: :t_n=\sum_^\infty \frac . Performing the summations, one gets: :t_n=1-\gamma + \sum_^n (-1)^k \left \frac - \frac \right/math> where \gamma is the
Euler–Mascheroni constant Euler's constant (sometimes also called the Euler–Mascheroni constant) is a mathematical constant usually denoted by the lowercase Greek letter gamma (). It is defined as the limiting difference between the harmonic series and the natural ...
. These t_n play the analog of the Stieltjes constants, but for the
falling factorial In mathematics, the falling factorial (sometimes called the descending factorial, falling sequential product, or lower factorial) is defined as the polynomial :\begin (x)_n = x^\underline &= \overbrace^ \\ &= \prod_^n(x-k+1) = \prod_^(x-k) \,. \ ...
expansion. By writing :a_n=t_n - \frac 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 sequence In mathematics, a Sheffer sequence or poweroid is a polynomial sequence, i.e., a sequence of polynomials in which the index of each polynomial equals its degree, satisfying conditions related to the umbral calculus in combinatorics. They are na ...
s of polynomials. This expansion of the Riemann zeta is investigated in the following references. The coefficients are decreasing as :\left(\frac\right)^e^ \cos\left(\sqrt-\frac\right) + \mathcal \left(\frac\right).


References


General references

*
A. Ya. Khinchin Aleksandr Yakovlevich Khinchin (russian: Алекса́ндр Я́ковлевич Хи́нчин, french: Alexandre Khintchine; July 19, 1894 – November 18, 1959) was a USSR, Soviet mathematician and one of the most significant contributors ...
, ''Continued Fractions'', 1935, English translation University of Chicago Press, 1961 ''(See section 15).'' * K. I. Babenko, ''On a Problem of Gauss'', Soviet Mathematical Doklady 19:136–140 (1978) * K. I. Babenko and S. P. Jur'ev, ''On the Discretization of a Problem of Gauss'', Soviet Mathematical Doklady 19:731–735 (1978). * A. Durner, ''On a Theorem of Gauss–Kuzmin–Lévy.'' Arch. Math. 58, 251–256, (1992). * 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).


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 Vepsta
The Bernoulli Operator, the Gauss–Kuzmin–Wirsing Operator, and the Riemann Zeta
(2004) (PDF)


External links

* * {{DEFAULTSORT:Gauss-Kuzmin-Wirsing operator Continued fractions Dynamical systems