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

TL;DR

This paper constructs sequences with small signed harmonic sums and applies these methods to create multiplicative functions with exceptionally small logarithmic partial sums, revealing new behaviors in harmonic analysis of integers.

Contribution

It introduces novel constructions of sequences and multiplicative functions with unusually small harmonic sums, combining probabilistic and deterministic methods.

Findings

Sequences with small signed harmonic sums for dense subsets of natural numbers.

Existence of multiplicative functions with logarithmic sums decaying faster than typical.

Infinitely many N where the partial sums are exponentially small in N^{1/3}.

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.