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

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 $\mathcal{A}\subset \mathbb{N}\mathrm{.}$ We apply these methods to further construct completely multiplicative functions $f:\mathbb{N}\to \{-1,1\}$ with unusually small logarithmic partial sums, that is, $$\sum _{n\le N}\frac{f(n)}{n}\ll \exp (-c_0\frac{N^{1\mathrm{/}3}}{(\log N{)}^{1\mathrm{/}3}})$$ holds for infinitely many $N\to \mathrm{\infty }$. 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.