Large values of exponential sums with multiplicative coefficients

Abstract

In 1977 Montgomery and Vaughan gave tight bounds for exponential sums of the form $\sum_{n\leq x}f(n)e(n\alpha)$ 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 $|\alpha-a/q|\leq 1/(qx)$, showing their results to be ``best-possible'' by observing that the first part of their bound is more-or-less attained when $f(n)=\chi(n), \alpha=\frac aq$ where $\chi$ is a primitive character mod $q$, and the second part when $f(p)=e(-\alpha p)$ for all large primes $p$. La Bret\`eche and Granville proved that when $\alpha$ 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 $|t|<\log x$; 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 $p\leq x$ for some bounded integer $h$. We also study exponential sums $\sum_{n\leq x, P^+(n)\leq y} f(n) e(n\alpha)$ restricted to $y$-smooth (or $y$-friable) integers $n$. We conjecture that this sum is $\ll \frac{\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 $\sum_{n\leq x} \frac{f(n)}{n} e(n\alpha)$. 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.