Exponential sums over Möbius convolutions with applications to partitions
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866915831718346752 |
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| author | Basak, Debmalya Robles, Nicolas Zaharescu, Alexandru |
| author_facet | Basak, Debmalya Robles, Nicolas Zaharescu, Alexandru |
| contents | We consider partitions $p_{w}(n)$ of a positive integer $n$ arising from the generating functions \[ \sum_{n=1}^\infty p_{w}(n) z^n = \prod_{m \in \mathbb{N}} (1-z^m)^{-w(m)}, \] where the weights $w(m)$ are Möbius convolutions. We establish an upper bound for $p_w(n)$ and, as a consequence, we obtain an asymptotic formula involving the number of odd and even partitions emerging from the weights. In order to achieve the desired bounds on the minor arcs resulting from the Hardy-Littlewood circle method, we establish bounds on exponential sums twisted by Möbius convolutions. Lastly, we provide an explicit formula relating the contributions from the major arcs with a sum over the zeros of the Riemann zeta-function. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_17435 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Exponential sums over Möbius convolutions with applications to partitions Basak, Debmalya Robles, Nicolas Zaharescu, Alexandru Number Theory We consider partitions $p_{w}(n)$ of a positive integer $n$ arising from the generating functions \[ \sum_{n=1}^\infty p_{w}(n) z^n = \prod_{m \in \mathbb{N}} (1-z^m)^{-w(m)}, \] where the weights $w(m)$ are Möbius convolutions. We establish an upper bound for $p_w(n)$ and, as a consequence, we obtain an asymptotic formula involving the number of odd and even partitions emerging from the weights. In order to achieve the desired bounds on the minor arcs resulting from the Hardy-Littlewood circle method, we establish bounds on exponential sums twisted by Möbius convolutions. Lastly, we provide an explicit formula relating the contributions from the major arcs with a sum over the zeros of the Riemann zeta-function. |
| title | Exponential sums over Möbius convolutions with applications to partitions |
| topic | Number Theory |
| url | https://arxiv.org/abs/2312.17435 |