The Chowla conjecture and Landau-Siegel zeroes
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866910968120868864 |
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| author | Jaskari, Mikko Sachpazis, Stelios |
| author_facet | Jaskari, Mikko Sachpazis, Stelios |
| contents | Let $k\geq 2$ be an integer and let $λ$ be the Liouville function. Given $k$ non-negative distinct integers $h_1,\ldots,h_k$, the Chowla conjecture claims that $\sum_{n\leq x}λ(n+h_1)\cdots λ(n+h_k)=o(x)$ as $x\to\infty$. An unconditional answer to this conjecture is yet to be found, and in this paper, we take a conditional approach towards it. More precisely, we establish a non-trivial bound for the sums $\sum_{n\leq x}λ(n+h_1)\cdots λ(n+h_k)$ under the existence of a Landau-Siegel zero for $x$ in an interval that depends on the modulus of the character whose Dirichlet series corresponds to the Landau-Siegel zero. Our work constitutes an improvement over the previous related results of Germán and Kátai, Chinis, and Tao and Teräväinen. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_10663 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The Chowla conjecture and Landau-Siegel zeroes Jaskari, Mikko Sachpazis, Stelios Number Theory 11N36, 11N37 Let $k\geq 2$ be an integer and let $λ$ be the Liouville function. Given $k$ non-negative distinct integers $h_1,\ldots,h_k$, the Chowla conjecture claims that $\sum_{n\leq x}λ(n+h_1)\cdots λ(n+h_k)=o(x)$ as $x\to\infty$. An unconditional answer to this conjecture is yet to be found, and in this paper, we take a conditional approach towards it. More precisely, we establish a non-trivial bound for the sums $\sum_{n\leq x}λ(n+h_1)\cdots λ(n+h_k)$ under the existence of a Landau-Siegel zero for $x$ in an interval that depends on the modulus of the character whose Dirichlet series corresponds to the Landau-Siegel zero. Our work constitutes an improvement over the previous related results of Germán and Kátai, Chinis, and Tao and Teräväinen. |
| title | The Chowla conjecture and Landau-Siegel zeroes |
| topic | Number Theory 11N36, 11N37 |
| url | https://arxiv.org/abs/2409.10663 |