Kempner-like harmonic series
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866911808104693760 |
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| author | Allouche, Jean-Paul Morin, Claude |
| author_facet | Allouche, Jean-Paul Morin, Claude |
| contents | Inspired by a question asked on the list {\tt mathfun}, we revisit {\em Kempner-like series}, i.e., harmonic sums $\sum' 1/n$ where the integers $n$ in the summation have ``restricted'' digits. First we give a short proof that $\lim_{k \to \infty}(\sum_{s_2(n) = k} 1/n) = 2 \log 2$, where $s_2(n)$ is the sum of the binary digits of the integer $n$. Then we propose two generalizations. One generalization addresses the case where $s_2(n)$ is replaced with $s_b(n)$, the sum of $b$-ary digits in base $b$: we prove that $\lim_{k \to \infty}\sum_{s_b(n) = k} 1/n = (2 \log b)/(b-1)$. The second generalization replaces the sum of digits in base $2$ with any block-counting function in base $2$, e.g., the function $a(n)$ of -- possibly overlapping -- $11$'s in the base-$2$ expansion of $n$, for which we obtain $\lim_{k \to \infty}\sum_{a(n) = k} 1/n = 4 \log 2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2305_18180 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Kempner-like harmonic series Allouche, Jean-Paul Morin, Claude Number Theory 11A63 11B85 40A05 68R15 05A05 Inspired by a question asked on the list {\tt mathfun}, we revisit {\em Kempner-like series}, i.e., harmonic sums $\sum' 1/n$ where the integers $n$ in the summation have ``restricted'' digits. First we give a short proof that $\lim_{k \to \infty}(\sum_{s_2(n) = k} 1/n) = 2 \log 2$, where $s_2(n)$ is the sum of the binary digits of the integer $n$. Then we propose two generalizations. One generalization addresses the case where $s_2(n)$ is replaced with $s_b(n)$, the sum of $b$-ary digits in base $b$: we prove that $\lim_{k \to \infty}\sum_{s_b(n) = k} 1/n = (2 \log b)/(b-1)$. The second generalization replaces the sum of digits in base $2$ with any block-counting function in base $2$, e.g., the function $a(n)$ of -- possibly overlapping -- $11$'s in the base-$2$ expansion of $n$, for which we obtain $\lim_{k \to \infty}\sum_{a(n) = k} 1/n = 4 \log 2$. |
| title | Kempner-like harmonic series |
| topic | Number Theory 11A63 11B85 40A05 68R15 05A05 |
| url | https://arxiv.org/abs/2305.18180 |