  Let L be the transfer operator associated with the Gauss' continued fraction
map, known also as the Gauss-Kuzmin-Wirsing operator, acting on the Banach
space. 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 supported
by 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 each
individual eigenvalue in the trace formulas.
