Large values of exponential sums with multiplicative coefficients

TL;DR

This paper advances bounds on exponential sums with multiplicative coefficients, especially on minor arcs, and explores sums over smooth integers, proposing conjectures and proving near-optimal bounds.

Contribution

It introduces new bounds for exponential sums on minor arcs, studies sums over smooth integers, and formulates conjectures with partial proofs, extending classical results.

Findings

Bounds on exponential sums on minor arcs are significantly smaller unless specific pretension conditions hold.

Conjectures on sums over smooth integers and logarithmically weighted sums are proposed, with partial bounds proved.

Various technical results on multiplicative functions are established for broader applications.

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.