Improved bounds for the two-point logarithmic Chowla conjecture
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866915645745004544 |
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| author | Pilatte, Cédric |
| author_facet | Pilatte, Cédric |
| contents | Let $λ$ be the Liouville function, defined as $λ(n) := (-1)^{Ω(n)}$ where $Ω(n)$ is the number of prime factors of $n$ with multiplicity. In 2021, Helfgott and Radziwiłł proved that $$\sum_{n\leq x} \frac{1}{n} λ(n) λ(n+1) \ll \frac{\log x}{(\log \log x)^{1/2}},$$improving earlier results by Tao and Teräväinen. We prove that $$\sum_{n\leq x} \frac{1}{n} λ(n) λ(n+1) \ll (\log x)^{1-c}$$for some absolute constant $c>0$. This appears to be best possible with current methods. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_19357 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Improved bounds for the two-point logarithmic Chowla conjecture Pilatte, Cédric Number Theory Combinatorics Let $λ$ be the Liouville function, defined as $λ(n) := (-1)^{Ω(n)}$ where $Ω(n)$ is the number of prime factors of $n$ with multiplicity. In 2021, Helfgott and Radziwiłł proved that $$\sum_{n\leq x} \frac{1}{n} λ(n) λ(n+1) \ll \frac{\log x}{(\log \log x)^{1/2}},$$improving earlier results by Tao and Teräväinen. We prove that $$\sum_{n\leq x} \frac{1}{n} λ(n) λ(n+1) \ll (\log x)^{1-c}$$for some absolute constant $c>0$. This appears to be best possible with current methods. |
| title | Improved bounds for the two-point logarithmic Chowla conjecture |
| topic | Number Theory Combinatorics |
| url | https://arxiv.org/abs/2310.19357 |