An average version of Cilleruelo's conjecture for families of $S_n$-polynomials over a number field
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866911772080865280 |
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| author | Viglino, Ilaria |
| author_facet | Viglino, Ilaria |
| contents | For $ f\in\mathbb{Z}[X] $ an irreducible polynomial of degree $ n $, the Cilleruelo's conjecture states that$$\log(\mbox{lcm}(f(1),\dots,f(M)))\sim(n-1)M\log M$$as $ M\rightarrow+\infty $, where $ \mbox{lcm}(f(1),\dots,f(M)) $ is the least common multiple of $f(1),\dots,f(M)$. It's well-known for $ n=1 $ as a consequence of Dirichlet's Theorem for primes in arithmetic progression, and it was proved by Cilleruelo for quadratic polynomials. Recently the conjecture was shown by Rudnick and Zehavi for a large family of polynomials of any degree. We want to investigate an average version of the conjecture for $S_n$-polynomials with integral coefficients over a fixed extension $K/\mathbb{Q}$ by considering the least common multiple of ideals of $\mathcal{O}_K$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_03999 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | An average version of Cilleruelo's conjecture for families of $S_n$-polynomials over a number field Viglino, Ilaria Number Theory For $ f\in\mathbb{Z}[X] $ an irreducible polynomial of degree $ n $, the Cilleruelo's conjecture states that$$\log(\mbox{lcm}(f(1),\dots,f(M)))\sim(n-1)M\log M$$as $ M\rightarrow+\infty $, where $ \mbox{lcm}(f(1),\dots,f(M)) $ is the least common multiple of $f(1),\dots,f(M)$. It's well-known for $ n=1 $ as a consequence of Dirichlet's Theorem for primes in arithmetic progression, and it was proved by Cilleruelo for quadratic polynomials. Recently the conjecture was shown by Rudnick and Zehavi for a large family of polynomials of any degree. We want to investigate an average version of the conjecture for $S_n$-polynomials with integral coefficients over a fixed extension $K/\mathbb{Q}$ by considering the least common multiple of ideals of $\mathcal{O}_K$. |
| title | An average version of Cilleruelo's conjecture for families of $S_n$-polynomials over a number field |
| topic | Number Theory |
| url | https://arxiv.org/abs/2402.03999 |