A logarithmic structure theorem for multiplicative functions with small partial sums

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