Exponential sums with multiplicative coefficients

Abstract

We provide estimates for the exponential sum (Formula Presented) where x and α are real numbers and f is a multiplicative function satisfying f ≤ 1. Our main focus is the class of functions f which are supported on the positive proportion of primes up to x in the sense that ∑p≤x f(p)/p≫ log log x. For such f we obtain rather sharp estimates for F(x, α) by extending earlier results of H. L. Montgomery and R. C. Vaughan. Our results provide a partial answer to a question posed by G. Tenenbaum concerning such estimates. © 1999 American Mathematical Society.