Recursive construction of a series converging to the eigenvalues of the Gauss-Kuzmin-Wirsing operator

Details Recursive construction of a series converging to the eigenvalues of the Gauss-Kuzmin-Wirsing operator Recursive construction of a series converging to the eigenvalues of the Gauss-Kuzmin-Wirsing operator

2010-04-11

arxiv.org/abs/1004.1783v3

Mathematics

Number Theory

16 pages, 1 figure. This version is not yet for submission! Some parts of the main conjecture can be proven (namely, that the

Scientific paper

Based on the technique previously developed by the author, we present a conjecture which claims that the reciprocal of the n-th largest (in absolute value) eigenvalue of the Gauss-Kuzmin-Wirsing operator is equal to the sum of a certain infinite series. This series is constructed recurrently. It consists of rational functions with integer coefficients in two variables X, Y, specialized at X=n and Y=2^n. This gives a strong evidence to the conjecture of Mayer and Roepstorff that eigenvalues have alternating sign. Further, a very similar recursion yields a series for the dominant eigenvalue of the Mayer-Ruelle operator.

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