On exponential sums with multiplicative coefficients

Let $f:\mathbb{N}\to z\in \mathbb{C}:\mathrm{\mid }z\mathrm{\mid }\le 1$ be multiplicative, and suppose that $\alpha$ has a rational approximation $a\mathrm{/}q$ with $\mathrm{\mid }\alpha -a\mathrm{/}q\mathrm{\mid }\le 1\mathrm{/}(qx)$ and $q\le x\mathrm{/}((\log x{)}^2(\log \log x{)}^3)$. We prove that $\mid {\sum }_{n\le x}f(n)e(n\alpha )\mid \le (\frac{4}{\sqrt{6}}+o(1))\frac{x}{\sqrt{q}}+O\left(\frac{x}{\log x}\right)$, whereas complete multiplicativity improves $4\mathrm{/}\sqrt{6}$ 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.