Abstract
We compute the adjoint of the Serre derivative map with respect to the Petersson scalar product by using existing tools of nearly holomorphic modular forms. The Fourier coefficients of a cusp form of integer weight k, constructed using this method, involve special values of certain shifted Dirichlet series associated with a given cusp form f of weight k+2. As application, we get an asymptotic bound for the special values of these shifted Dirichlet series and also relate these special values with the Fourier coefficients of f. We also give a formula for the Ramanujan tau function in terms of the special values of the shifted Dirichlet series associated to the Ramanujan delta function.
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