Assuming the Generalized Riemann Hypothesis (GRH), we show using the asymptotic large sieve that 91% of the zeros of primitive Dirichlet L-functions are simple. This improves on earlier work ofÖzlük which gives a proportion of at most 86%. We further compute q-analogue of the Pair Correlation Function F (α) averaged over all primitive Dirichlet L-functions in the range |α| < 2 . Previously such a result was available only when the average included all the characters χ. As a corollary of our results, we obtain an asymptotic formula for a sum over characters similar to the one encountered in the Barban-Davenport-Halberstam Theorem.

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Abstract. Let q be a prime and −D < −4 be an odd fundamental discriminant such that q splits in Q( √ −D). For f a weight zero Hecke-Maass newform of level q and Θ χ the weight one theta series of level D corresponding to an ideal class group character χ of Q( √ −D), we establish a hybrid subconvexity bound for L(f ×Θ χ , s) at s = 1/2 when q ≍ D η for 0 < η < 1. With this circle of ideas, we show that the Heegner points of level q and discriminant D become equidistributed, in a natural sense, as q, D → ∞ for q ≤ D 1/20−ε . Our approach to these problems is connected to estimating the L 2 -restriction norm of a Maass form of large level q when restricted to the collection of Heegner points. We furthermore establish bounds for quadratic twists of Hecke-Maass L-functions with simultaneously large level and large quadratic twist, and hybrid bounds for quadratic Dirichlet L-functions in certain ranges.