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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2604.06848 |
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| _version_ | 1866918434585968640 |
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| author | Klurman, Oleksiy Mangerel, Alexander P. |
| author_facet | Klurman, Oleksiy Mangerel, Alexander P. |
| contents | We establish an asymptotic formula for the logarithmic mean value of a 1-bounded multiplicative function that is sharp in many cases of interest. We derive from it a variety of applications, making progress on several old problems.
As a first application, we show that if $f$ is a completely multiplicative function taking values in $[-1,1]$ then there is a constant $c > 0$ such that for every $x \geq 3$, $$ L_f(x) := \sum_{n \leq x} \frac{f(n)}{n} > -\frac{c}{(\log x)^{1-2/π}}, $$ thus significantly improving on a 20-year-old result of Granville and Soundararajan. We also show that the exponent of $\log x$ in this result can be improved to $-1+o(1)$, as long as $f$ does not ``behave like'' the Liouville function $λ$ in a precise sense.
As a second application, we show that for a Rademacher random completely multiplicative function $\mathbf{f}$, the probability that $L_{\mathbf{f}}(x)$ is negative is $O(\exp(-x^c))$ for some $c \in (0,1)$, thus establishing a previously conjectured bound.
Finally, we obtain a converse theorem for small absolute values $|L_f(x)|$, and construct examples $f$ that show that it is (essentially) best possible. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_06848 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A Halász-type asymptotic formula for logarithmic means and its consequences Klurman, Oleksiy Mangerel, Alexander P. Number Theory Probability We establish an asymptotic formula for the logarithmic mean value of a 1-bounded multiplicative function that is sharp in many cases of interest. We derive from it a variety of applications, making progress on several old problems. As a first application, we show that if $f$ is a completely multiplicative function taking values in $[-1,1]$ then there is a constant $c > 0$ such that for every $x \geq 3$, $$ L_f(x) := \sum_{n \leq x} \frac{f(n)}{n} > -\frac{c}{(\log x)^{1-2/π}}, $$ thus significantly improving on a 20-year-old result of Granville and Soundararajan. We also show that the exponent of $\log x$ in this result can be improved to $-1+o(1)$, as long as $f$ does not ``behave like'' the Liouville function $λ$ in a precise sense. As a second application, we show that for a Rademacher random completely multiplicative function $\mathbf{f}$, the probability that $L_{\mathbf{f}}(x)$ is negative is $O(\exp(-x^c))$ for some $c \in (0,1)$, thus establishing a previously conjectured bound. Finally, we obtain a converse theorem for small absolute values $|L_f(x)|$, and construct examples $f$ that show that it is (essentially) best possible. |
| title | A Halász-type asymptotic formula for logarithmic means and its consequences |
| topic | Number Theory Probability |
| url | https://arxiv.org/abs/2604.06848 |