Résumé
This thesis establishes some strong asymptotic formulae for the harmonic mollified second moment of a family of Rankin-Selberg $L$-functions. The main contribution is a substancial improvement of the admissible length of the mollifier which is done by solving a shifted convolution problem by a spectral method on average. A first consequence is a new sharp subconvexity bound for Rankin-Selberg L-functions in the level aspect which has many already known arithmetic applications. Moreover, infinitely many Rankin-Selberg L-functions having at most eight non-trivial real zeros are produced and some new non-trivial estimates of the analytic rank of the family studied are obtained.