Logarithmic exponential sums with multiplicative coefficients

Source abstract

Let $f$ be a $1$-bounded multiplicative function, and suppose that $|α-a/q|\leq 1/(qx)$, where $(a,q)=1$ and $q\leq x$. We prove that $$\left|\sum_{n\leq x}\frac{f(n)}{n}e( nα)\right| \leq \left(\frac{4}{\sqrt{6}}+o(1)\right)\frac{\log x}{\sqrt{q}}+O(\log q).$$ Here the constant $4/\sqrt{6}$ is optimal. This settles two conjectures of Granville and Lamzouri recently. If $f$ is completely multiplicative, the constant improves to $1$. The main new ingredient is a two-prime Ramaré decomposition identity.