Real zeros and size of Rankin-Selberg \(L\)-functions in the level aspect (Q2501390)

This work is motivated by that of \textit{J. B. Conrey} and \textit{K. Soundararajan} [Invent. Math. 150, No. 1, 1--44 (2002; Zbl 1042.11053)], who showed that a positive proportion of quadratic Dirichlet \(L\)-functions have only trivial zeros on the real line. The author fixes a primitive form \(g\) (satisfying certain conditions), and considers the \(L\)-series attached to the Rankin-Selberg convolution \(f\times g\), where \(f\) runs over a certain class of cusp forms, depending on \(g\). It is then shown that the \(L\)-function has at most 8 nontrivial real zeros, for a positive proportion of such \(f\). The proof depends on establishing asymptotic formulae for the mean-square (with respect to \(f)\) of mollified versions of the \(L\)-function. The paper shows that one can use substantially longer mollifiers than was previously possible.