Large values of exponential sums with multiplicative coefficients

In 1977 Montgomery and Vaughan gave tight bounds for exponential sums of the form ${\sum }_{n\le x}f(n)e(n\alpha )$ where $f$ is a $1$-bounded multiplicative function and $\alpha \in \mathbb{R}$, close to the conjectured $\ll \frac{x}{\sqrt{q}}+\frac{x}{\log x}$ where $\alpha$ is best approximated by $\mathrm{\mid }\alpha -a\mathrm{/}q\mathrm{\mid }\le 1\mathrm{/}(qx)$, showing their results to be best-possible'' by observing that the first part of their bound is more-or-less attained when $f(n)=χ(n), α=\frac aq$ where $χ$ is a primitive character mod $q$, and the second part when $f(p)=e(-αp)$ for all large primes $p$. La Bretèche and Granville proved that when $α$ lies on a major arc the exponential sum is significantly smaller unless $f$ pretends to be’’ $\chi (n)n^{it}$ for some character $\chi$ and real number $\mathrm{\mid }t\mathrm{\mid }<\log x$; and herein we prove that when $\alpha$ lies on a minor arc, the exponential sum is significantly smaller unless $f(p)$ pretends to be $e(-hp\alpha )$ for primes $p\le x$ for some bounded integer $h$. We also study exponential sums ${\sum }_{n\le x,P^{+}(n)\le y}f(n)e(n\alpha )$ restricted to $y$-smooth (or $y$-friable) integers $n$. We conjecture that this sum is $\ll \frac{\mathrm{\Psi }(x,y)}{\sqrt{q}}+\frac{\sqrt{xy}}{\log x}$ in a wide range of parameters, show that if true this is best possible, and prove an upper bound in a wide range that is only slightly weaker than the conjecture. Finally we study the logarithmically weighted exponential sums ${\sum }_{n\le x}\frac{f(n)}{n}e(n\alpha )$. We conjecture that this sum is $\ll \frac{\log x}{\sqrt{q}}+\log q$ in a wide range of parameters, show that if true this is best possible, and prove an upper bound in a wide range that is only slightly weaker than the conjecture. Along the way, we will prove various technical results about multiplicative functions which may be of use elsewhere.