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Real zeros and size of Rankin-Selberg \(L\)-functions in the level aspect. (English) Zbl 1122.11059
This work is motivated by that of J. B. Conrey and 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.

MSC:
11M41 Other Dirichlet series and zeta functions
11F66 Langlands \(L\)-functions; one variable Dirichlet series and functional equations
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