An upper bound for the number of smooth values of a polynomial and its applications
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866918156017074176 |
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| author | Mine, Masahiro |
| author_facet | Mine, Masahiro |
| contents | We prove a new upper bound for the number of smooth values of a polynomial with integer coefficients. This improves Timofeev's previous result unless the polynomial is a product of linear polynomials with integer coefficients. As an application, we provide another proof for a result of Cassels which was used to prove that the Hurwitz zeta-function with algebraic irrational parameter has infinitely many zeros on the domain of convergence. We also apply the main result to a problem on primitive divisors of quadratic polynomials. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_09558 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | An upper bound for the number of smooth values of a polynomial and its applications Mine, Masahiro Number Theory Primary 11N32, Secondary 11N25 We prove a new upper bound for the number of smooth values of a polynomial with integer coefficients. This improves Timofeev's previous result unless the polynomial is a product of linear polynomials with integer coefficients. As an application, we provide another proof for a result of Cassels which was used to prove that the Hurwitz zeta-function with algebraic irrational parameter has infinitely many zeros on the domain of convergence. We also apply the main result to a problem on primitive divisors of quadratic polynomials. |
| title | An upper bound for the number of smooth values of a polynomial and its applications |
| topic | Number Theory Primary 11N32, Secondary 11N25 |
| url | https://arxiv.org/abs/2410.09558 |