On a problem of Erdős and Ingham
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866908720091365376 |
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| author | Yip, Fredy |
| author_facet | Yip, Fredy |
| contents | We give a short and elementary argument answering a question of Erdős and Ingham negatively. Erdős and Ingham showed that a Tauberian estimate they considered was equivalent to the non-vanishing of $1+\sum_{k}a_k^{-1-it}$ for any real number $t$ and any sequence $1<a_1<a_2<\cdots$ of positive integers such that $\sum_k a_k^{-1}<\infty$. We disprove this statement. In fact, we show that for any complex number $λ$ and any non-zero real number $t$, there exists a sequence $1<a_1<a_2<\cdots$ of positive integers such that $\sum_k a_k^{-1}<\infty$ and $\sum_k a_k^{-1-it} = λ$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_16528 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On a problem of Erdős and Ingham Yip, Fredy Classical Analysis and ODEs Number Theory We give a short and elementary argument answering a question of Erdős and Ingham negatively. Erdős and Ingham showed that a Tauberian estimate they considered was equivalent to the non-vanishing of $1+\sum_{k}a_k^{-1-it}$ for any real number $t$ and any sequence $1<a_1<a_2<\cdots$ of positive integers such that $\sum_k a_k^{-1}<\infty$. We disprove this statement. In fact, we show that for any complex number $λ$ and any non-zero real number $t$, there exists a sequence $1<a_1<a_2<\cdots$ of positive integers such that $\sum_k a_k^{-1}<\infty$ and $\sum_k a_k^{-1-it} = λ$. |
| title | On a problem of Erdős and Ingham |
| topic | Classical Analysis and ODEs Number Theory |
| url | https://arxiv.org/abs/2512.16528 |