Logarithmic exponential sums with multiplicative coefficients

Abstract

Let $f$ be a $1$-bounded multiplicative function, and suppose that $|\alpha -a/q|\le 1/(qx)$, where $(a,q)=1$ and $q\le x$. We prove that $$\left|\sum _{n\le x}\frac{f(n)}{n}e(n\alpha )\right|\le \left(\frac{4}{\sqrt{6}}+o(1)\right)\frac{\log x}{\sqrt{q}}+O(\log q).$$ 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.

arXiv comment13 pages