A folklore conjecture on multiplicative exponential sums

Abstract

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

28 pages