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| Format: | Preprint |
| Published: |
2019
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/1910.09969 |
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| _version_ | 1866913959094779904 |
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| author | Lowry-Duda, David |
| author_facet | Lowry-Duda, David |
| contents | We study the general theory of weighted Dirichlet series and associated summatory functions of their coefficients. We show that any non-real pole leads to oscillatory error terms. This applies even if there are infinitely many non-real poles with the same real part. Further, we consider the case when the non-real poles lie near, but not on, a line. The method of proof is a generalization of classical ideas applied to study the oscillatory behavior of the error term in the prime number theorem. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1910_09969 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | Non-real Poles and Irregularity of Distribution Lowry-Duda, David Number Theory Complex Variables We study the general theory of weighted Dirichlet series and associated summatory functions of their coefficients. We show that any non-real pole leads to oscillatory error terms. This applies even if there are infinitely many non-real poles with the same real part. Further, we consider the case when the non-real poles lie near, but not on, a line. The method of proof is a generalization of classical ideas applied to study the oscillatory behavior of the error term in the prime number theorem. |
| title | Non-real Poles and Irregularity of Distribution |
| topic | Number Theory Complex Variables |
| url | https://arxiv.org/abs/1910.09969 |