- Gauss–Kuzmin–Wirsing operator
:"GKW" redirects here. For the Indian engineering firm see
Guest Keen Williams .In
mathematics , the Gauss–Kuzmin–Wirsing operator occurs in the study ofcontinued fractions ; it is also related to theRiemann zeta function .Introduction
The Gauss–Kuzmin–Wirsing operator is the
transfer operator of the Gauss map:h(x)=1/x-lfloor 1/x floor.,
This operator acts on functions as
:Gf] (x) = sum_{n=1}^infty frac {1}{(x+n)^2} f left(frac {1}{x+n} ight).
The zeroth
eigenfunction of this operator is:frac 1{ln 2} frac 1{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 theGauss-Kuzmin distribution . This follows in part because the Gauss map acts as a truncatingshift operator for thecontinued fraction s: 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 "λ"1 = −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 .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}-sint_0^1 h(x) x^{s-1} ; dx
which implies that
:zeta(s)=frac{s}{s-1}-sint_0^1 x left [Gx^{s-1} ight] , 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 ight] .
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)^n {n choose k} left [ frac{1}{k} + frac {zeta(k+1)} {k+1} ight]
where gamma is the
Euler-Mascheroni constant . These t_n play the analog of theStieltjes constants , but for thefalling factorial expansion. By writing:a_n=t_n - frac{1}{2(n+1)}
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 ofSheffer sequence s of polynomials.This expansion of the Riemann zeta is investigated in [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] [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] [Luis Báez-Duarte, " [http://arxiv.org/abs/math.NT/0307215 A New Necessary and Sufficient Condition for the Riemann Hypothesis] " (2003) "ArXiv" math.NT/0307215] [Luis Báez-Duarte, "A sequential Riesz-like criterion for the Riemann hypothesis", "Internation Journal of Mathematics and Mathematical Sciences", 21, pp. 3527-3537 (2005)] [Philippe Flajolet and Linas Vepstas, " [http://arxiv.org/abs/math.CA/0611332 On differences of zeta values] ", "ArXiv" math.CA/0611332 to appear in "Journal of Computational and Applied Mathematics" (2007)] The coefficients are decreasing as
:left(frac{2n}{pi} ight)^{1/4}e^{-sqrt{4pi n cosleft(sqrt{4pi n}-frac{5pi}{8} ight) +mathcal{O} left(frac{e^{-sqrt{4pi n}{n^{1/4 ight).
References
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
* Keith Briggs, " [http://keithbriggs.info/documents/wirsing.pdf A precise computation of the Gauss-Kuzmin-Wirsing constant] " (2003) "(Contains a very extensive collection of references.)"
* A. Durner, "On a Theorem of Gauss-Kuzmin-Lévy." Arch. Math. 58, 251-256, (1992). MR 93c:11056
* Phillipe Flajolet andBrigitte Vallée , " [http://pauillac.inria.fr/algo/flajolet/Publications/gauss-kuzmin.ps On the Gauss-Kuzmin-Wirsing Constant] " (1995).
* 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
* Linas Vepstas [http://www.linas.org/math/gkw.pdf The Bernoulli Operator, the Gauss-Kuzmin-Wirsing Operator, and the Riemann Zeta] (2004) (PDF)External links
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