6.5: A Formula of Gauss, a Theorem of Kuzmin and Levi and a Problem of Arnold
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 8857
In this connection Gauss asked about a probability \(c_k\) for a number \(k\) to appear as an element of a continued fraction. Such a probability is defined in a natural way: as a limit when \(N \rightarrow \infty\) of the number of occurrences of \(k\) among the first \(N\) elements of the continued fraction enpension. Moreover, Gauss provided an answer, but never published the proof. Two different proofs were found independently by R.O.Kuzmin (1928) and P. Lévy (1929) (see for a detailed exposition of the R.O.Kuzmin’s proof).
[Gauss] For almost every real \(\alpha\) the probability for a number \(k\) to appear as an element in the continued fraction expansion of \(\alpha\) is \[\label{ga} c_k=\frac{1}{\ln 2} \ln \left( 1 + \frac{1}{k(k+2)} \right).\]
Remarks. 1. The words "for almost every \(\alpha\)" mean that the measure of the set of exceptions is zero.
2. Even the existence of \(p_k\) (defined as a limit) is highly nontrivial.
Theorem [Gauss] may (and probably should) be considered as a result from ergodic theory rather than number theory. This constructs a bridge between these two areas of Mathematics and explains the recent attention to continued fractions of the mathematicians who study dynamical systems. In particular, V.I.Arnold formulated the following open problem. Consider the set of pairs of integers \((a,b)\) such that the corresponding points on the plane are contained in a quarter of a circle of radii \(N\): \[a^2 + b^2 \leq N^2.\] Expand the numbers \(p/q\) into continued fractions and compute the frequencies \(s_k\) for the appearance of \(k\) in these fractions. Do these frequencies have limits as \(N \rightarrow \infty\)? If so, do these limits have anything to do with the probabilities, given by ([ga])? These questions demand nothing but experimental computer investigation, and such an experiment may be undertaken by a student. Of course, it would be extremely challenging to find a phenomena experimentally in this way and to prove it after that theoretically.
Of course, one can consider more general kinds of continued fractions. In particular, one may ease the assumption that the elements are positive integers and consider, allowing arbitrary reals as the elements (the question of convergence may usually be solved). The following identities were discovered independently by three prominent mathematicians. The English mathematician R.J. Rogers found and proved these identities in 1894, Ramanujan found the identities (without proof) and formulated them in his letter to Hardy from India in 1913. Independently, being separated from England by the war, I. J. Schur found the identities and published two different proofs in 1917. We refer an interested reader to for a detailed discussion and just state the amazing identities here. \[[0;e^{2\pi},e^{4\pi},e^{6\pi},e^{8\pi}, \ldots ]= \left(\sqrt{\frac{5+\sqrt{5}}{2}}  \frac{\sqrt{5}+1}{2} \right) e^{2\pi/5}\]
\[[1;e^{\pi},e^{2\pi},e^{3\pi},e^{4\pi}, \ldots ]= \left(\sqrt{\frac{5\sqrt{5}}{2}}  \frac{\sqrt{5}1}{2} \right) e^{\pi/5}\]
Exercises

Prove that \(c_k\) really define a probability distribution, namely that \[\sum_{k=1}^\infty c_k =1.\]
Contributors
Dr. Wissam Raji, Ph.D., of the American University in Beirut. His work was selected by the Saylor Foundation’s Open Textbook Challenge for public release under a Creative Commons Attribution (CC BY) license.