On some Exponential Sums

Abstract

Let f be a multiplicative function, and let α be an irrational number. In this paper we want to estimate the exponential sum $\sum\nolimits_{n \leqslant x} {f\left( n \right)e\left( {na} \right)}$. If f is the constant multiplicative function 1 then trivially

$$ \sum\limits_{n \leqslant x} {1\left( n \right)e\left( {na} \right)\, = \,\sum\limits_{n \leqslant x} {e\left( {na} \right)\, = \,o\left( x \right);} } $$

in fact, the sum is bounded in this case.