Small values of signed harmonic sums and logarithmic means of multiplicative functions
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866913094389727232 |
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| author | Klurman, Oleksiy Munsch, Marc Sun, Yu-Chen |
| author_facet | Klurman, Oleksiy Munsch, Marc Sun, Yu-Chen |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_04694 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Small values of signed harmonic sums and logarithmic means of multiplicative functions Klurman, Oleksiy Munsch, Marc Sun, Yu-Chen Number Theory Combinatorics Probability Primary 11N56, Secondary 11B99, 11M20 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. |
| title | Small values of signed harmonic sums and logarithmic means of multiplicative functions |
| topic | Number Theory Combinatorics Probability Primary 11N56, Secondary 11B99, 11M20 |
| url | https://arxiv.org/abs/2605.04694 |