Exponential sums with multiplicative coefficients without the Ramanujan conjecture

Abstract

We study the exponential sum involving multiplicative function f under milder conditions on the range of f, which generalizes the work of Montgomery and Vaughan. As an application, we prove cancellation in the sum of additively twisted coefficients of automorphic L-function on $$\text {GL}_m$$ $$(m\ge 4)$$ , uniformly in the additive character.