A logarithmic structure theorem for multiplicative functions with small partial sums

Abstract

Let $D\in \mathbb{N}$, let $A>D+1$, and let $Q⩾3$. Consider the class of multiplicative functions $f:\mathbb{N}\to \mathbb{C}$ such that $|{\sum }_{n⩽x}f(n)|\le x(\log Q{)}^{A-D-1}/(\log x{)}^A$ for all $x⩾Q$, and such that $|\hat{I}{›}_f|⩽D\hat{I}›$, where $\hat{I}{›}_f$ is defined via the Dirichlet convolution identity $f\log =\hat{I}{›}_f\ast f$ and $\hat{I}›$ denotes von Mangoldt's function. We prove there exist parameters $m\in \{0,1,\dots ,D\}$ and $Q=Q_D⩽Q_{D-1}\le \cdots ⩽Q_m<Q_{m+1}=\infty$ such that ${\sum }_{p\in I}\mathrm{Re}(f(p)+j)/p=O_{A,D}(1)$ for all $j=m,m+1,\dots ,D$ and all compact intervals $I\subset [Q_{j+1},Q_j)$. Moreover, when $|{\sum }_{n⩽x}f(n)|\le x^{1-1/\log Q}/(\log x{)}^{D+1}$ for all $x⩾Q$, we relate the parameters $m$ and $Q_j$ to the location of zeroes of the Dirichlet series ${\sum }_{n⩾1}f(n)/n^s$ in the ball $B(1,1/\log Q)$. These results generalize work of the author when $D=1$. Their proof builds on earlier work of the author with Soundararajan, and of Sachpazis.

18 pages