We provide a new expansion of the Fourier coefficient of the perturbing function of thePCR3Body problem in terms of Hansen coefficients. This gives us a precise asymptotic formulafor the coefficient in the region of application of KAM theory (i. e., small value of eccentricity and semimajoraxis. See, e. g., [17]). Moreover, in the above region, we study the presence of zeros of the Fourier coefficient for coprime modes \((m,k)\in\mathbb{Z}^{2}\) and the presence of common zeros as functions of actions between coefficients relative to modes \((m,k)\) , \((2m,2k)\) and \((m,k)\) , \((2m,2k)\) , \((3m,3k)\) .Thanks to the previous expansion, this numerical analysis is done up to order \(60\) in the powerof eccentricity and semimajor axis. This is thefirst step for a possible application of [4, 9] to the PCR3Body Problem that would imply a reduction in terms of measure in the phase space of the so-called “non-torus” set from \(O(1-\sqrt{\varepsilon})\) (implied by standard KAM theory) to \(O(1-\varepsilon|\log\varepsilon|^{c})\) for some \(c>0\) .
