Liouville function, von Mangoldt function and norm forms at random binary forms
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908416484573184 |
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| author | Diao, Yijie |
| author_facet | Diao, Yijie |
| contents | We analyze the average behavior of various arithmetic functions at the values of degree $d$ binary forms ordered by height, with probability $1$. This approach yields averaged versions of the Chowla conjecture and the Bateman-Horn conjecture for random binary forms. Furthermore, we show that the rational Hasse principle holds for almost all Châtelet varieties defined by a fixed norm form of degree $e$ and by varying binary forms of fixed degree $d$, provided $e$ divides $d$. This proves an average version of a conjecture of Colliot-Thélène. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_18065 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Liouville function, von Mangoldt function and norm forms at random binary forms Diao, Yijie Number Theory 11N32 (11N37, 11D57, 11G35) We analyze the average behavior of various arithmetic functions at the values of degree $d$ binary forms ordered by height, with probability $1$. This approach yields averaged versions of the Chowla conjecture and the Bateman-Horn conjecture for random binary forms. Furthermore, we show that the rational Hasse principle holds for almost all Châtelet varieties defined by a fixed norm form of degree $e$ and by varying binary forms of fixed degree $d$, provided $e$ divides $d$. This proves an average version of a conjecture of Colliot-Thélène. |
| title | Liouville function, von Mangoldt function and norm forms at random binary forms |
| topic | Number Theory 11N32 (11N37, 11D57, 11G35) |
| url | https://arxiv.org/abs/2506.18065 |