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| :''"GKW" redirects here. For the Indian engineering firm see [[Guest Keen Williams]].
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| In [[mathematics]], the '''Gauss–Kuzmin–Wirsing operator''', named after [[Carl Gauss]], [[Rodion Kuzmin|Rodion Osievich Kuzmin]] and [[Eduard Wirsing]], occurs in the study of [[continued fractions]]; it is also related to the [[Riemann zeta function]].
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| ==Introduction==
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| The Gauss–Kuzmin–Wirsing operator is the [[transfer operator]] of the Gauss map
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| :<math>h(x)=1/x-\lfloor 1/x \rfloor.\,</math>
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| This operator acts on functions as
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| :<math>[Gf](x) = \sum_{n=1}^\infty \frac {1}{(x+n)^2} f \left(\frac {1}{x+n}\right).</math>
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| The zeroth [[eigenfunction]] of this operator is
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| :<math>\frac 1{\ln 2}\ \frac 1{1+x}</math>
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| 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 fraction]]s: if
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| : <math>x=[0;a_1,a_2,a_3,\dots]\,</math>
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| is the continued fraction representation of a number 0 < ''x'' < 1, then
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| : <math>h(x)=[0;a_2,a_3,\dots].\,</math>
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| Additional eigenvalues can be computed numerically; the next eigenvalue is ''λ''<sub>1</sub> = −0.3036630029... {{OEIS|A038517}}
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| 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]].
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| ==Relationship to the Riemann zeta==
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| The GKW operator is related to the [[Riemann zeta function]]. Note that the zeta can be written as
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| :<math>\zeta(s)=\frac{1}{s-1}-s\int_0^1 h(x) x^{s-1} \; dx</math>
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| which implies that
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| :<math>\zeta(s)=\frac{s}{s-1}-s\int_0^1 x \left[Gx^{s-1} \right]\, dx </math>
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| by change-of-variable.
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| ==Matrix elements==
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| Consider the [[Taylor series]] expansions at x=1 for a function ''f''(''x'') and <math>g(x)=[Gf](x)</math>. That is, let
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| :<math>f(1-x)=\sum_{n=0}^\infty (-x)^n \frac{f^{(n)}(1)}{n!}</math>
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| 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
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| :<math>(-1)^m \frac{g^{(m)}(1)}{m!} = \sum_{n=0}^\infty G_{mn} (-1)^n \frac{f^{(n)}(1)}{n!},</math>
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| where the matrix elements of the GKW operator are given by
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| :<math>G_{mn}=\sum_{k=0}^n (-1)^k {n \choose k} {k+m+1 \choose m} \left[ \zeta (k+m+2)- 1\right].</math>
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| 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.
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| ==Riemann zeta==
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| The Riemann zeta can be written as
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| :<math>\zeta(s)=\frac{s}{s-1}-s \sum_{n=0}^\infty (-1)^n {s-1 \choose n} t_n</math>
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| where the <math>t_n</math> are given by the matrix elements above:
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| :<math>t_n=\sum_{m=0}^\infty \frac{G_{mn}} {(m+1)(m+2)}.</math>
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| Performing the summations, one gets:
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| :<math>t_n=1-\gamma + \sum_{k=1}^n (-1)^k {n \choose k} \left[ \frac{1}{k} - \frac {\zeta(k+1)} {k+1} \right]</math>
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| where <math>\gamma</math> is the [[Euler–Mascheroni constant]]. These <math>t_n</math> play the analog of the [[Stieltjes constants]], but for the [[falling factorial]] expansion. By writing
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| :<math>a_n=t_n - \frac{1}{2(n+1)}</math>
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| one gets: ''a''<sub>0</sub> = −0.0772156... and ''a''<sub>1</sub> = −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]]s of polynomials.
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| This expansion of the Riemann zeta is investigated in <ref>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</ref><ref>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</ref><ref>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</ref><ref>Luis Báez-Duarte, "A sequential Riesz-like criterion for the Riemann hypothesis", ''Internation Journal of Mathematics and Mathematical Sciences'', '''21''', pp. 3527–3537 (2005)</ref><ref>Philippe Flajolet and Linas Vepstas, "[http://arxiv.org/abs/math.CA/0611332 On differences of zeta values]", J. Comput. Appl. Math. 220, No. 1-2, 58-73 (2008).</ref> The coefficients are decreasing as
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| :<math>\left(\frac{2n}{\pi}\right)^{1/4}e^{-\sqrt{4\pi n}}
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| \cos\left(\sqrt{4\pi n}-\frac{5\pi}{8}\right) +
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| \mathcal{O} \left(\frac{e^{-\sqrt{4\pi n}}}{n^{1/4}}\right).</math>
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| ==References==
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| <references/>
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| ==General references==
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| * [[Aleksandr Khinchin|A. Ya. Khinchin]], ''Continued Fractions'', 1935, English translation University of Chicago Press, 1961 ISBN 0-486-69630-8 ''(See section 15).''
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| * K. I. Babenko, ''On a Problem of Gauss'', Soviet Mathematical Doklady '''19''':136–140 (1978) MR 57 #12436
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| * 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
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| * A. Durner, ''On a Theorem of Gauss–Kuzmin–Lévy.'' Arch. Math. '''58''', 251–256, (1992). MR 93c:11056
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| * A. J. MacLeod, ''High-Accuracy Numerical Values of the Gauss–Kuzmin Continued Fraction Problem.'' Computers Math. Appl. '''26''', 37–44, (1993).
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| * 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
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| ==Further reading==
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| * Giedrius Alkauskas, ''[http://arxiv.org/pdf/1210.4083v3.pdf Transfer operator for the Gauss' continued fraction map. I. Structure of the eigenvalues and trace formulas]'' (2013).
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| * 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.)''
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| * Phillipe Flajolet and [[Brigitte Vallée]], ''[http://pauillac.inria.fr/algo/flajolet/Publications/gauss-kuzmin.ps On the Gauss–Kuzmin–Wirsing Constant]'' (1995).
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| * Linas Vepstas [http://www.linas.org/math/gkw.pdf The Bernoulli Operator, the Gauss–Kuzmin–Wirsing Operator, and the Riemann Zeta] (2004) (PDF)
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| == External links ==
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| * {{MathWorld|urlname=Gauss-Kuzmin-WirsingConstant|title=Gauss-Kuzmin-Wirsing Constant}}
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| * {{SloanesRef|sequencenumber=A038517}}
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| {{DEFAULTSORT:Gauss-Kuzmin-Wirsing operator}}
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| [[Category:Continued fractions]]
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| [[Category:Dynamical systems]]
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