Bounded exponential sums with multiplicative coefficients

Abstract We investigate when the exponential sum is bounded, for a multiplicative function and . We show that under natural assumptions, is bounded only when is very close to a twisted Dirichlet character . We obtain sharper classification results for functions that are completely multiplicative or take only finitely many values, including a complete classification in the case when is completely multiplicative and is irrational. We also prove a stronger classification under the assumption that the sum is bounded for a positive measure set of .