Modified Dirichlet character sums over the $k$-free integers
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866913809458790400 |
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| author | Bueno, Caio |
| author_facet | Bueno, Caio |
| contents | The main question of this paper is the following: how much cancellation can the partial sums restricted to the $k$-free integers up to $x$ of a $\pm 1$ multiplicative function $f$ be in terms of $x$? Building upon the recent paper by Q. Liu, Acta Math. Sin. (Engl. Ser.) 39 (2023), no. 12, 2316-2328, we prove that under the Riemann Hypothesis for quadratic Dirichlet $L$-functions, we can get $x^{1/(k+1)}$ cancellation when $f$ is a modified quadratic Dirichlet character, i.e., $f$ is completely multiplicative and for some quadratic Dirichlet character $χ$, $f(p)=χ(p)$ for all but a finite subset of prime numbers. This improves the conditional results by Aymone, Medeiros and the author cf. Ramanujan J. 59 (2022), no. 3, 713-728. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2411_08268 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Modified Dirichlet character sums over the $k$-free integers Bueno, Caio Number Theory 11M06, 11N37, 11L40 The main question of this paper is the following: how much cancellation can the partial sums restricted to the $k$-free integers up to $x$ of a $\pm 1$ multiplicative function $f$ be in terms of $x$? Building upon the recent paper by Q. Liu, Acta Math. Sin. (Engl. Ser.) 39 (2023), no. 12, 2316-2328, we prove that under the Riemann Hypothesis for quadratic Dirichlet $L$-functions, we can get $x^{1/(k+1)}$ cancellation when $f$ is a modified quadratic Dirichlet character, i.e., $f$ is completely multiplicative and for some quadratic Dirichlet character $χ$, $f(p)=χ(p)$ for all but a finite subset of prime numbers. This improves the conditional results by Aymone, Medeiros and the author cf. Ramanujan J. 59 (2022), no. 3, 713-728. |
| title | Modified Dirichlet character sums over the $k$-free integers |
| topic | Number Theory 11M06, 11N37, 11L40 |
| url | https://arxiv.org/abs/2411.08268 |