A folklore conjecture on multiplicative exponential sums

We prove a folklore conjecture on exponential sums with multiplicative coefficients: if $f$ is multiplicative with $\mathrm{\mid }f(n)\mathrm{\mid }\le 1$, $(a,q)=1$, $\mathrm{\mid }\alpha -a\mathrm{/}q\mathrm{\mid }\le q^{-2}$ and $q\le N\mathrm{/}(\log N{)}^2$, then $$\sum _{n\le N}f(n)\mathrm{e}(n\alpha )\ll \frac{N}{\log N}+\frac{N}{\sqrt{q}}\mathrm{.}$$ Complete multiplicativity is not required. We also prove two related conjectured bounds: the refinement in which $q$ is replaced by $q(1+N\mathrm{\mid }\alpha -a\mathrm{/}q\mathrm{\mid })$, when this quantity is at most $N\mathrm{/}(\log N{)}^2$, and the corresponding bound for the sum with logarithmic weights. Finally, we show that the bound can fail for multiplicative $f$ satisfying only $\mathrm{\mid }f(p)\mathrm{\mid }\le A$ at primes and ${\sum }_{n\le x}\mathrm{\mid }f(n){\mathrm{\mid }}^2\le A^{2}x$, for every fixed $A>1$.