On exponential sums with multiplicative coefficients

Abstract

Let $f:\mathbb{N}\to z\in \mathbb{C}:|z|\le 1$ be multiplicative, and suppose that $\alpha$ has a rational approximation $a/q$ with $|\alpha -a/q|\le 1/(qx)$ and $q\le x/((\log x{)}^2(\log \log x{)}^3)$. We prove that $\left|{\sum }_{n\le x}f(n)e(n\alpha )\right|\le \left(\frac{4}{\sqrt{6}}+o(1)\right)\frac{x}{\sqrt{q}}+O\left(\frac{x}{\log x}\right)$, whereas complete multiplicativity improves $4/\sqrt{6}$ to $1$. Both constants are optimal. In particular, this confirms a folklore conjecture on exponential sums with multiplicative coefficients motivated by the work of Montgomery and Vaughan (1977). We also obtain logarithmically weighted estimates and give counterexamples to related conjectures.

arXiv comment27 pages