Effective estimates for exponential sums with multiplicative coefficients

Let $f$ be multiplicative, with $\mathrm{\mid }f(p)\mathrm{\mid }\le A$ at primes and ${\sum }_{n\le x}\mathrm{\mid }f(n){\mathrm{\mid }}^2\le A^{2}x$ for every $x\ge 1$. If $\mathrm{\mid }\alpha -a\mathrm{/}q\mathrm{\mid }\le q^{-2}$, $(a,q)=1$, and $3\le R\le q\le N\mathrm{/}R$, we prove $$\sum _{n\le N}f(n)\mathrm{e}(n\alpha ){\ll }_A\frac{N}{\log N}+\frac{N}{\sqrt{R}}\sqrt{\log \log (3R)}$$ with effective implied constants. Montgomery and Vaughan proved this with second term $N{R}^{-1\mathrm{/}2}(\log R{)}^{3\mathrm{/}2}$, and, for $1$-bounded functions, Bachman replaced it by $N{R}^{-1\mathrm{/}2}\sqrt{\log R\log \log R}$. We remove the factor $\sqrt{\log R}$ from Bachman’s second term while retaining the original coefficient hypotheses of Montgomery and Vaughan. A more precise estimate records the distance from a rational number. The proof combines the Brun-Titchmarsh inequality on short intervals with maximal Fourier estimates derived from the Carleson-Hunt theorem; the local bounds permit arbitrary prime-dependent prefixes. We also prove sharpness of the square-root displacement dependence.