Some identities related to Riemann zeta-function
Abstract It is well known that the Riemann zeta-function ζ ( s ) $\zeta(s)$ plays a very important role in the study of analytic number theory. In this paper, we use the elementary method and some new inequalities to study the computational problem of one kind of reciprocal sums related to the Riema...
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| Format: | Article |
| Language: | English |
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SpringerOpen
2016-01-01
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| Series: | Journal of Inequalities and Applications |
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| Online Access: | http://link.springer.com/article/10.1186/s13660-016-0980-9 |
| _version_ | 1811193904621420544 |
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| author | Lin Xin |
| author_facet | Lin Xin |
| author_sort | Lin Xin |
| collection | DOAJ |
| description | Abstract It is well known that the Riemann zeta-function ζ ( s ) $\zeta(s)$ plays a very important role in the study of analytic number theory. In this paper, we use the elementary method and some new inequalities to study the computational problem of one kind of reciprocal sums related to the Riemann zeta-function at the integer point s ≥ 2 $s\geq2$ , and for the special values s = 2 , 3 $s=2, 3$ , we give two exact identities for the integer part of the reciprocal sums of the Riemann zeta-function. For general integer s ≥ 4 $s\geq4$ , we also propose an interesting open problem. |
| first_indexed | 2024-04-12T00:17:14Z |
| format | Article |
| id | doaj.art-9ec24e1e897942c1848508e80cd718ef |
| institution | Directory Open Access Journal |
| issn | 1029-242X |
| language | English |
| last_indexed | 2024-04-12T00:17:14Z |
| publishDate | 2016-01-01 |
| publisher | SpringerOpen |
| record_format | Article |
| series | Journal of Inequalities and Applications |
| spelling | doaj.art-9ec24e1e897942c1848508e80cd718ef2022-12-22T03:55:49ZengSpringerOpenJournal of Inequalities and Applications1029-242X2016-01-01201611610.1186/s13660-016-0980-9Some identities related to Riemann zeta-functionLin Xin0School of Mathematics, Northwest UniversityAbstract It is well known that the Riemann zeta-function ζ ( s ) $\zeta(s)$ plays a very important role in the study of analytic number theory. In this paper, we use the elementary method and some new inequalities to study the computational problem of one kind of reciprocal sums related to the Riemann zeta-function at the integer point s ≥ 2 $s\geq2$ , and for the special values s = 2 , 3 $s=2, 3$ , we give two exact identities for the integer part of the reciprocal sums of the Riemann zeta-function. For general integer s ≥ 4 $s\geq4$ , we also propose an interesting open problem.http://link.springer.com/article/10.1186/s13660-016-0980-9Riemann zeta-functioninequalityfunction [ x ] $[x]$identityelementary method |
| spellingShingle | Lin Xin Some identities related to Riemann zeta-function Journal of Inequalities and Applications Riemann zeta-function inequality function [ x ] $[x]$ identity elementary method |
| title | Some identities related to Riemann zeta-function |
| title_full | Some identities related to Riemann zeta-function |
| title_fullStr | Some identities related to Riemann zeta-function |
| title_full_unstemmed | Some identities related to Riemann zeta-function |
| title_short | Some identities related to Riemann zeta-function |
| title_sort | some identities related to riemann zeta function |
| topic | Riemann zeta-function inequality function [ x ] $[x]$ identity elementary method |
| url | http://link.springer.com/article/10.1186/s13660-016-0980-9 |
| work_keys_str_mv | AT linxin someidentitiesrelatedtoriemannzetafunction |