Mean Value of <i>r</i>-gcd-sum and <i>r</i>-lcm-Sum Functions
In this paper we perform a further investigation for <i>r</i>-gcd-sum and <i>r</i>-lcm-sum functions. By making use of the properties of generalization of Euler’s <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">...
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| Format: | Article |
| Language: | English |
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MDPI AG
2022-10-01
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| Series: | Symmetry |
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| Online Access: | https://www.mdpi.com/2073-8994/14/10/2080 |
| _version_ | 1797469827624861696 |
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| author | Zhengjin Bu Zhefeng Xu |
| author_facet | Zhengjin Bu Zhefeng Xu |
| author_sort | Zhengjin Bu |
| collection | DOAJ |
| description | In this paper we perform a further investigation for <i>r</i>-gcd-sum and <i>r</i>-lcm-sum functions. By making use of the properties of generalization of Euler’s <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>φ</mi></semantics></math></inline-formula>-function, Abel’s identity and elementary arguments, we derive asymptotic formulas for the average of the <i>r</i>-gcd-sum function, <i>r</i>-lcm-sum function and their generalizations. Moreover, we also study the sums of reciprocals of <i>r</i>-gcd and <i>r</i>-lcm. |
| first_indexed | 2024-03-09T19:26:34Z |
| format | Article |
| id | doaj.art-26ff7397a7414f87846b903fc4cef211 |
| institution | Directory Open Access Journal |
| issn | 2073-8994 |
| language | English |
| last_indexed | 2024-03-09T19:26:34Z |
| publishDate | 2022-10-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Symmetry |
| spelling | doaj.art-26ff7397a7414f87846b903fc4cef2112023-11-24T02:52:00ZengMDPI AGSymmetry2073-89942022-10-011410208010.3390/sym14102080Mean Value of <i>r</i>-gcd-sum and <i>r</i>-lcm-Sum FunctionsZhengjin Bu0Zhefeng Xu1Research Center for Number Theory and Its Applications, Northwest University, Xi’an 710127, ChinaResearch Center for Number Theory and Its Applications, Northwest University, Xi’an 710127, ChinaIn this paper we perform a further investigation for <i>r</i>-gcd-sum and <i>r</i>-lcm-sum functions. By making use of the properties of generalization of Euler’s <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>φ</mi></semantics></math></inline-formula>-function, Abel’s identity and elementary arguments, we derive asymptotic formulas for the average of the <i>r</i>-gcd-sum function, <i>r</i>-lcm-sum function and their generalizations. Moreover, we also study the sums of reciprocals of <i>r</i>-gcd and <i>r</i>-lcm.https://www.mdpi.com/2073-8994/14/10/2080r-gcd-sum functionr-lcm-sum functionmean valueasymptotic formulas |
| spellingShingle | Zhengjin Bu Zhefeng Xu Mean Value of <i>r</i>-gcd-sum and <i>r</i>-lcm-Sum Functions Symmetry r-gcd-sum function r-lcm-sum function mean value asymptotic formulas |
| title | Mean Value of <i>r</i>-gcd-sum and <i>r</i>-lcm-Sum Functions |
| title_full | Mean Value of <i>r</i>-gcd-sum and <i>r</i>-lcm-Sum Functions |
| title_fullStr | Mean Value of <i>r</i>-gcd-sum and <i>r</i>-lcm-Sum Functions |
| title_full_unstemmed | Mean Value of <i>r</i>-gcd-sum and <i>r</i>-lcm-Sum Functions |
| title_short | Mean Value of <i>r</i>-gcd-sum and <i>r</i>-lcm-Sum Functions |
| title_sort | mean value of i r i gcd sum and i r i lcm sum functions |
| topic | r-gcd-sum function r-lcm-sum function mean value asymptotic formulas |
| url | https://www.mdpi.com/2073-8994/14/10/2080 |
| work_keys_str_mv | AT zhengjinbu meanvalueofirigcdsumandirilcmsumfunctions AT zhefengxu meanvalueofirigcdsumandirilcmsumfunctions |