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Let L be the transfer operator associated with the Gauss' continued fractionmap, known also as the Gauss-Kuzmin-Wirsing operator, acting on the Banachspace. In this work we prove an asymptotic formula for the eigenvalues of L.This settles, in a stronger form, the conjectures of D. Mayer and G. Roepstorff(1988), A.J. MacLeod (1992), Ph. Flajolet and B. Vallee (1995), also supportedby several other authors. Further, we find an exact series for the eigenvalues,which also gives the canonical decomposition of trace formulas due to D. Mayer(1976) and K.I. Babenko (1978). This crystallizes the contribution of eachindividual eigenvalue in the trace formulas.

Transfer operator for the Gauss' continued fraction map. I. Structure of the eigenvalues and trace formulas