Exponential Sums with Additive Coefficients and its Consequences to Weighted Partitions
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arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866912172905332736 |
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| author | Das, Madhuparna |
| author_facet | Das, Madhuparna |
| contents | In this article, we consider the weighted partition function $p_f(n)$ given by the generating series $\sum_{n=1}^{\infty} p_f(n)z^n = \prod_{n\in\mathbb{N}^{*}}(1-z^n)^{-f(n)}$, where we restrict the class of weight functions to strongly additive functions. Originally proposed in a paper by Yang, this problem was further examined by Debruyne and Tenenbaum for weight functions taking positive integer values. We establish an asymptotic formula for this generating series in a broader context, which notably can be used for the class of multiplicative functions. Moreover, we employ a classical result by Montgomery-Vaughan to estimate exponential sums with additive coefficients, supported on minor arcs. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_21129 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Exponential Sums with Additive Coefficients and its Consequences to Weighted Partitions Das, Madhuparna Number Theory 11N60, 11L15, 11P84 In this article, we consider the weighted partition function $p_f(n)$ given by the generating series $\sum_{n=1}^{\infty} p_f(n)z^n = \prod_{n\in\mathbb{N}^{*}}(1-z^n)^{-f(n)}$, where we restrict the class of weight functions to strongly additive functions. Originally proposed in a paper by Yang, this problem was further examined by Debruyne and Tenenbaum for weight functions taking positive integer values. We establish an asymptotic formula for this generating series in a broader context, which notably can be used for the class of multiplicative functions. Moreover, we employ a classical result by Montgomery-Vaughan to estimate exponential sums with additive coefficients, supported on minor arcs. |
| title | Exponential Sums with Additive Coefficients and its Consequences to Weighted Partitions |
| topic | Number Theory 11N60, 11L15, 11P84 |
| url | https://arxiv.org/abs/2412.21129 |