The moments of split greatest common divisors
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866916358209404928 |
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| author | Jha, Abhishek Nath, Ayan Tron, Emanuele |
| author_facet | Jha, Abhishek Nath, Ayan Tron, Emanuele |
| contents | Sequences of the form $(\gcd(u_n,v_n))_{n \in \mathbb N}$, with $(u_n)_n$, $(v_n)_n$ sums of $S$-units, have been considered by several authors. The study of $\gcd(n,u_n)$ corresponds, following Silverman, to divisibility sequences arising from the split algebraic group $\mathbb G_{\mathrm{a}} \times \mathbb G_{\mathrm{m}}$; in this case, Sanna determined all asymptotic moments of the arithmetic function $\log\,\gcd (n,u_n)$ when $(u_n)_n$ is a Lucas sequence. Here, we characterize the asymptotic behavior of the moments themselves $\sum_{n \leq x}\,\gcd(n,u_n)^λ$, thus solving the moment problem for $\mathbb G_{\mathrm{a}} \times \mathbb G_{\mathrm{m}}$. We give both unconditional and conditional results, the latter only relying on standard conjectures in analytic number theory. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2408_05820 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The moments of split greatest common divisors Jha, Abhishek Nath, Ayan Tron, Emanuele Number Theory 11N56, 11B37 Sequences of the form $(\gcd(u_n,v_n))_{n \in \mathbb N}$, with $(u_n)_n$, $(v_n)_n$ sums of $S$-units, have been considered by several authors. The study of $\gcd(n,u_n)$ corresponds, following Silverman, to divisibility sequences arising from the split algebraic group $\mathbb G_{\mathrm{a}} \times \mathbb G_{\mathrm{m}}$; in this case, Sanna determined all asymptotic moments of the arithmetic function $\log\,\gcd (n,u_n)$ when $(u_n)_n$ is a Lucas sequence. Here, we characterize the asymptotic behavior of the moments themselves $\sum_{n \leq x}\,\gcd(n,u_n)^λ$, thus solving the moment problem for $\mathbb G_{\mathrm{a}} \times \mathbb G_{\mathrm{m}}$. We give both unconditional and conditional results, the latter only relying on standard conjectures in analytic number theory. |
| title | The moments of split greatest common divisors |
| topic | Number Theory 11N56, 11B37 |
| url | https://arxiv.org/abs/2408.05820 |