Small values of signed harmonic sums and logarithmic means of multiplicative functions

Abstract

We construct sequences $\{a_n{\}}_{n\in \mathbb{N}}\in \{-1,1{\}}^{\mathbb{N}}$ with small values of signed harmonic sums \[ \sum_{n\in\mathcal{A}\cap[1,N]}\frac{a_n}{n}, \] for any reasonably dense subsets $\mathcal{A}\subset \mathbb{N}.$ We apply these methods to further construct completely multiplicative functions $f:\mathbb{N}\to \{-1,1\}$ with unusually small logarithmic partial sums, that is, \[ \sum_{n \leq N}\frac{f(n)}{n} \ll \exp\left(-c_0 \frac{N^{1/3}}{(\log N)^{1/3}} \right) \] holds for infinitely many $N\to \infty$. The proofs combine careful analysis of the small-scale distribution of random harmonic sums over subsets of $\mathbb{N}$, together with deterministic inductive arguments inspired by the ``anatomy" of integers.