We extend to an arbitrary number field the best known bounds towards Ramanujan for the group GLn, n = 2, 3, 4. In particular, we present a technique which overcomes the analytic obstacles posed by the presence of an infinite group of units.

We establish zero-free regions tapering as an inverse power of the analytic conductor for Rankin-Selberg L-functions on GL n ×GL n ′ . Such zero-free regions are equivalent to commensurate lower bounds on the edge of the critical strip, and in the case of L(s, π × π), on the residue at s = 1. As an application we show that a cuspidal automorphic representation on GL n is determined by a finite number of its Dirichlet series coefficients, and that this number grows at most polynomially in the analytic conductor.

Abstract. We discuss progress towards the Ramanujan conjecture for the group GL n and its relation to various other topics in analytic number theory.

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We specify sufficient conditions for the square modulus of the local parameters of a family of GL n cusp forms to be bounded on average. These conditions are global in nature and are satisfied for n 4. As an application, we show that Rankin-Selberg L-functions on GL n 1 × GL n 2 , for n i 4, satisfy the standard convexity bound.