Analytic number theory and families of automorphic L-functions

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Sub-Weyl subconvexity for Dirichlet $L$-functions to prime power moduli (2014, background): “Proving a subconvex estimate for any given L-function requires deep arithmetic considerations and can have important arithmetic, geometric, or dynamical consequences; see surveys [13, 16].”
Weak subconvexity for central values of $L$-functions (2008, methodology): “In the context of symmetric square L-functions, see Luo [24], and Cogdell and Michel [4]. use the convexity bound to establish that (see Lemma 4.2 below)7 (1.7) ∑ n≤x aπ(n) ≪ x log x , provided x ≥ C 12 (logC)B for some positive constant B.”