A simple proof of generalizations of number-theoretic sums
For positive integers 𝑘, 𝑚, and 𝑛, let 𝑆𝑘 𝑚(𝑛) be the sum of all elements in the finite set {𝑥 𝑘 : 1 ≤ 𝑥 ≤ 𝑛⁄𝑚 , (𝑥, 𝑛) = 1}. The formula for 𝑆𝑘 𝑚(𝑛) is established and simpler formulae for 𝑆𝑘 𝑚(𝑛) under some conditions on 𝑚 and 𝑛 are verified. The explicit formulae for 𝑆1 2 𝑎 (𝑛) and 𝑆2...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
Prince of Songkla University
2022-04-01
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| Series: | Songklanakarin Journal of Science and Technology (SJST) |
| Subjects: | |
| Online Access: | https://rdo.psu.ac.th/sjst/journal/44-2/22.pdf |
| _version_ | 1811243012901044224 |
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| author | Yanapat Tongron Narakorn Rompurk Kanasri |
| author_facet | Yanapat Tongron Narakorn Rompurk Kanasri |
| author_sort | Yanapat Tongron |
| collection | DOAJ |
| description | For positive integers 𝑘, 𝑚, and 𝑛, let 𝑆𝑘
𝑚(𝑛) be the sum of all elements in the finite set {𝑥
𝑘
: 1 ≤ 𝑥 ≤ 𝑛⁄𝑚 , (𝑥, 𝑛) = 1}.
The formula for 𝑆𝑘
𝑚(𝑛) is established and simpler formulae for 𝑆𝑘
𝑚(𝑛) under some conditions on 𝑚 and 𝑛 are verified. The explicit
formulae for 𝑆1
2
𝑎
(𝑛) and 𝑆2
2
𝑎
(𝑛), where 2
𝑎
|𝑛 and 𝑎 ≥ 1, are also provided. |
| first_indexed | 2024-04-12T13:59:53Z |
| format | Article |
| id | doaj.art-8863d723529544afa4ba7555e3ea076a |
| institution | Directory Open Access Journal |
| issn | 0125-3395 |
| language | English |
| last_indexed | 2024-04-12T13:59:53Z |
| publishDate | 2022-04-01 |
| publisher | Prince of Songkla University |
| record_format | Article |
| series | Songklanakarin Journal of Science and Technology (SJST) |
| spelling | doaj.art-8863d723529544afa4ba7555e3ea076a2022-12-22T03:30:14ZengPrince of Songkla UniversitySongklanakarin Journal of Science and Technology (SJST)0125-33952022-04-0144245045810.14456/sjst-psu.2022.62A simple proof of generalizations of number-theoretic sumsYanapat Tongron0Narakorn Rompurk Kanasri1Mathematics and Applied Statistics Program, Faculty of Science and Technology, Nakhon Ratchasima Rajabhat University, Mueang, Nakhon Ratchasima, 30000 ThailandDepartment of Mathematics, Faculty of Science, Khon Kaen University, Mueang, Khon Kaen, 40002 ThailandFor positive integers 𝑘, 𝑚, and 𝑛, let 𝑆𝑘 𝑚(𝑛) be the sum of all elements in the finite set {𝑥 𝑘 : 1 ≤ 𝑥 ≤ 𝑛⁄𝑚 , (𝑥, 𝑛) = 1}. The formula for 𝑆𝑘 𝑚(𝑛) is established and simpler formulae for 𝑆𝑘 𝑚(𝑛) under some conditions on 𝑚 and 𝑛 are verified. The explicit formulae for 𝑆1 2 𝑎 (𝑛) and 𝑆2 2 𝑎 (𝑛), where 2 𝑎 |𝑛 and 𝑎 ≥ 1, are also provided.https://rdo.psu.ac.th/sjst/journal/44-2/22.pdfarithmetic functioneuler’s phi-functionmöbius functionmöbius inversion formula |
| spellingShingle | Yanapat Tongron Narakorn Rompurk Kanasri A simple proof of generalizations of number-theoretic sums Songklanakarin Journal of Science and Technology (SJST) arithmetic function euler’s phi-function möbius function möbius inversion formula |
| title | A simple proof of generalizations of number-theoretic sums |
| title_full | A simple proof of generalizations of number-theoretic sums |
| title_fullStr | A simple proof of generalizations of number-theoretic sums |
| title_full_unstemmed | A simple proof of generalizations of number-theoretic sums |
| title_short | A simple proof of generalizations of number-theoretic sums |
| title_sort | simple proof of generalizations of number theoretic sums |
| topic | arithmetic function euler’s phi-function möbius function möbius inversion formula |
| url | https://rdo.psu.ac.th/sjst/journal/44-2/22.pdf |
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