On Nonzero Coefficients of Binary Cyclotomic Polynomials
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866911130626031616 |
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| author | Shparlinski, Igor E. Wijaya, Laurence P. |
| author_facet | Shparlinski, Igor E. Wijaya, Laurence P. |
| contents | Let $\vartheta(m)$ is number of nonzero coefficients in the $m$-th cyclotomic polynomial. For real $γ> 0$ and $x \ge 2$ we define $$H_γ(x)=\#\left\{m:~m=pq \le x, \ p<q\text{ primes }, \ \vartheta(m)\le m^{1/2+γ}\right\}, $$ and show that for any fixed $η> 0$, uniformly over $γ$ with $$9/20+η\le γ\le 1/2 -η, $$ we have an asymptotic formula $$ H_γ(x)\sim C(γ)x^{1/2+γ}/ \log x, \qquad x \to \infty, $$ where $C(γ)> 0$ is an explicit constant depending only on $γ$. This extends the previous result of {É}.~Fouvry (2013), which has $12/25$ instead of $9/20$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_07229 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On Nonzero Coefficients of Binary Cyclotomic Polynomials Shparlinski, Igor E. Wijaya, Laurence P. Number Theory 11B83, 11L07 Let $\vartheta(m)$ is number of nonzero coefficients in the $m$-th cyclotomic polynomial. For real $γ> 0$ and $x \ge 2$ we define $$H_γ(x)=\#\left\{m:~m=pq \le x, \ p<q\text{ primes }, \ \vartheta(m)\le m^{1/2+γ}\right\}, $$ and show that for any fixed $η> 0$, uniformly over $γ$ with $$9/20+η\le γ\le 1/2 -η, $$ we have an asymptotic formula $$ H_γ(x)\sim C(γ)x^{1/2+γ}/ \log x, \qquad x \to \infty, $$ where $C(γ)> 0$ is an explicit constant depending only on $γ$. This extends the previous result of {É}.~Fouvry (2013), which has $12/25$ instead of $9/20$. |
| title | On Nonzero Coefficients of Binary Cyclotomic Polynomials |
| topic | Number Theory 11B83, 11L07 |
| url | https://arxiv.org/abs/2307.07229 |