Mathematics Colloquium

L-functions are generalizations of the Riemann zeta function. Their analytic properties control the asymptotic behavior of prime numbers in various refined senses. Conjecturally, every L-function is a "standard L-function" arising from an automorphic form. A problem of recurring interest, with widespread applications, has been to establish nontrivial bounds for L-functions. I will survey some recent results addressing this problem. The proofs involve the analysis of integrals of automorphic forms, approached through the lens of representation theory. I will emphasize the role played by the orbit method, developed in a quantitative form along the lines of microlocal analysis.