Effective estimates for exponential sums with multiplicative coefficients

Abstract

Let $f$ be multiplicative, with $|f(p)|\le A$ at primes and ${\sum }_{n\le x}|f(n){|}^2\le A^{2}x$ for every $x\ge 1$. If $|\alpha -a/q|\le q^{-2}$, $(a,q)=1$, 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 $N{R}^{-1/2}(\log R{)}^{3/2}$, and, for $1$-bounded functions, Bachman replaced it by $N{R}^{-1/2}\sqrt{\log R\log \log R}$. 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.

21 pages