On the asymptotics of Kempner-Irwin sums
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866909991103889408 |
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| author | Burnol, Jean-François |
| author_facet | Burnol, Jean-François |
| contents | Let $I(b,d,k)$ be the subseries of the harmonic series keeping the integers having exactly $k$ occurrences of the digit $d$ in base $b$. We prove the existence of an asymptotic expansion to all orders in descending powers of $b$, for fixed $d$ and $k$, of $I(b,d,k)-b\log(b)$. We explicitly give, depending on cases, either four or five terms. The coefficients involve the values of the zeta function at the integers. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2404_13763 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the asymptotics of Kempner-Irwin sums Burnol, Jean-François Number Theory Classical Analysis and ODEs 11Y60, 11M06 (Primary) 11A63, 44A60, 30C10, 41A60 (Secondary) Let $I(b,d,k)$ be the subseries of the harmonic series keeping the integers having exactly $k$ occurrences of the digit $d$ in base $b$. We prove the existence of an asymptotic expansion to all orders in descending powers of $b$, for fixed $d$ and $k$, of $I(b,d,k)-b\log(b)$. We explicitly give, depending on cases, either four or five terms. The coefficients involve the values of the zeta function at the integers. |
| title | On the asymptotics of Kempner-Irwin sums |
| topic | Number Theory Classical Analysis and ODEs 11Y60, 11M06 (Primary) 11A63, 44A60, 30C10, 41A60 (Secondary) |
| url | https://arxiv.org/abs/2404.13763 |