First moments of Rankin–Selberg convolutions of automorphic forms on GL(2)

We obtain a first moment formula for Rankin–Selberg convolution L-series of holomorphic modular forms or Maass forms of arbitrary level on GL(2), with an orthonormal basis of Maass forms. One consequence is the best result to date, uniform in level, spectral value and weight, for the equality of two Maass or holomorphic cusp forms if their Rankin–Selberg convolutions with the orthonormal basis of Maass forms 𝑢𝑗 is equal at the center of the critical strip for sufficiently many 𝑢𝑗. The main novelty of our approach is the new way the error terms are treated. They are brought into an exact form that provides optimal estimates for this first moment case, and also provide a basis for an extension to second moments, which will appear in another work.