Effective estimates for exponential sums with multiplicative coefficients

Abstract: Let $f$ be multiplicative, with $|f(p)|\le A$ at primes and $\sum_{n\le x}|f(n)|^2\le A<sup>2x$ for every $x\ge1$. If $|α-a/q|\le q<sup>{-2}$, $(a,q)=1$, and $3\le R\le q\le N/R$, we prove [ \sum_{n\le N}f(n)\operatorname{e}(nα) \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 $NR^{-1/2}(\log R)<sup>{3/2}$, and, for $1$-bounded functions, Bachman replaced it by $NR^{{-1/2}\sqrt{\log} R\log\log R}$. 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.