A logarithmic structure theorem for multiplicative functions with small partial sums

Abstract

Let $D\in\mathbb{N}$, let $A>D+1$, and let $Q\geqslant3$. Consider the class of multiplicative functions $f:\mathbb{N}\to\mathbb{C}$ such that $|\sum_{n\leqslant x}f(n)|\le x(\log Q)^{A-D-1}/(\log x)^A$ for all $x\geqslant Q$, and such that $|\Lambda_f|\leqslant D\Lambda$, where $\Lambda_f$ is defined via the Dirichlet convolution identity $f\log=\Lambda_f*f$ and $\Lambda$ denotes von Mangoldt's function. We prove there exist parameters $m\in\{0,1,\dots,D\}$ and $Q=Q_D\leqslant Q_{D-1}\le \cdots\leqslant Q_m<Q_{m+1}=\infty$ 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 $|\sum_{n\leqslant x}f(n)|\le x^{1-1/\log Q}/(\log x)^{D+1}$ for all $x\geqslant Q$, we relate the parameters $m$ and $Q_j$ to the location of zeroes of the Dirichlet series $\sum_{n\geqslant1} f(n)/n^s$ 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.