A logarithmic structure theorem for multiplicative functions with small partial sums

TL;DR

This paper establishes a logarithmic structure theorem for multiplicative functions with small partial sums, revealing their behavior and zero distribution of associated Dirichlet series under certain bounds.

Contribution

It generalizes previous results by characterizing the structure of multiplicative functions with small partial sums for arbitrary D, linking parameters to zeroes of Dirichlet series.

Findings

Existence of parameters m and Q_j that control the behavior of f on primes.

Bounded sums over primes in specific intervals for functions with small partial sums.

Connection between partial sum bounds and zeroes of Dirichlet series in a specified ball.

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.