On values of the Riemann zeta function at positive integers
We give new proofs of some known results on the values of the Riemann zeta function at positive integers and obtain some new theorems related to these values. Considering even zeta values as ζ(2 n ) = η n π 2 n , we obtain the generating functions of the sequences η n and (−1) n η n . Using the Rie...
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| Veröffentlicht in: | Lithuanian mathematical journal 2020, Vol.60 (1), p.9-24 |
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| container_title | Lithuanian mathematical journal |
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| creator | Dil, Ayhan Boyadzhiev, Khristo N. Aliev, Ilham A. |
| description | We give new proofs of some known results on the values of the Riemann zeta function at positive integers and obtain some new theorems related to these values. Considering even zeta values as ζ(2
n
) =
η
n
π
2
n
, we obtain the generating functions of the sequences
η
n
and (−1)
n
η
n
. Using the Riemann–Lebesgue lemma, we give recurrence relations for ζ(2
n
) and ζ(2
n
+ 1). Furthermore, we prove some series equations for
∑
k
=
1
∞
−
1
k
−
1
ζ
p
+
k
/
k
. |
| doi_str_mv | 10.1007/s10986-019-09456-7 |
| format | Article |
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n
) =
η
n
π
2
n
, we obtain the generating functions of the sequences
η
n
and (−1)
n
η
n
. Using the Riemann–Lebesgue lemma, we give recurrence relations for ζ(2
n
) and ζ(2
n
+ 1). Furthermore, we prove some series equations for
∑
k
=
1
∞
−
1
k
−
1
ζ
p
+
k
/
k
.</description><identifier>ISSN: 0363-1672</identifier><identifier>EISSN: 1573-8825</identifier><identifier>DOI: 10.1007/s10986-019-09456-7</identifier><language>eng</language><publisher>New York: Springer US</publisher><subject>Actuarial Sciences ; Integers ; Mathematics ; Mathematics and Statistics ; Number Theory ; Numbers ; Ordinary Differential Equations ; Probability Theory and Stochastic Processes ; Sequences</subject><ispartof>Lithuanian mathematical journal, 2020, Vol.60 (1), p.9-24</ispartof><rights>Springer Science+Business Media, LLC, part of Springer Nature 2019</rights><rights>Springer Science+Business Media, LLC, part of Springer Nature 2019.</rights><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c319t-f749af8e694f37077d200fd39d1a1f1f16e4bbf0adcd2481b279df0ca2b515f33</citedby><cites>FETCH-LOGICAL-c319t-f749af8e694f37077d200fd39d1a1f1f16e4bbf0adcd2481b279df0ca2b515f33</cites></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktopdf>$$Uhttps://link.springer.com/content/pdf/10.1007/s10986-019-09456-7$$EPDF$$P50$$Gspringer$$H</linktopdf><linktohtml>$$Uhttps://link.springer.com/10.1007/s10986-019-09456-7$$EHTML$$P50$$Gspringer$$H</linktohtml><link.rule.ids>314,776,780,27903,27904,41467,42536,51297</link.rule.ids></links><search><creatorcontrib>Dil, Ayhan</creatorcontrib><creatorcontrib>Boyadzhiev, Khristo N.</creatorcontrib><creatorcontrib>Aliev, Ilham A.</creatorcontrib><title>On values of the Riemann zeta function at positive integers</title><title>Lithuanian mathematical journal</title><addtitle>Lith Math J</addtitle><description>We give new proofs of some known results on the values of the Riemann zeta function at positive integers and obtain some new theorems related to these values. Considering even zeta values as ζ(2
n
) =
η
n
π
2
n
, we obtain the generating functions of the sequences
η
n
and (−1)
n
η
n
. Using the Riemann–Lebesgue lemma, we give recurrence relations for ζ(2
n
) and ζ(2
n
+ 1). Furthermore, we prove some series equations for
∑
k
=
1
∞
−
1
k
−
1
ζ
p
+
k
/
k
.</description><subject>Actuarial Sciences</subject><subject>Integers</subject><subject>Mathematics</subject><subject>Mathematics and Statistics</subject><subject>Number Theory</subject><subject>Numbers</subject><subject>Ordinary Differential Equations</subject><subject>Probability Theory and Stochastic Processes</subject><subject>Sequences</subject><issn>0363-1672</issn><issn>1573-8825</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2020</creationdate><recordtype>article</recordtype><recordid>eNp9kMtKAzEUhoMoWKsv4CrgOnqSzOSCKyneoFAQXYfMTKJT2kxNMgV9eqMjuJOzOJvvP5cPoXMKlxRAXiUKWgkCVBPQVS2IPEAzWktOlGL1IZoBF5xQIdkxOklpDVB4CjN0vQp4bzejS3jwOL85_NS7rQ0Bf7pssR9Dm_shYJvxbkh97vcO9yG7VxfTKTrydpPc2W-fo5e72-fFA1mu7h8XN0vScqoz8bLS1isndOW5BCk7BuA7rjtqqS8lXNU0HmzXdqxStGFSdx5ay5qa1p7zObqY5u7i8F4uzWY9jDGUlYZxVfNKKCkKxSaqjUNK0Xmzi_3Wxg9DwXxLMpMkUySZH0lGlhCfQqnAoTz1N_qf1BeMRWm8</recordid><startdate>2020</startdate><enddate>2020</enddate><creator>Dil, Ayhan</creator><creator>Boyadzhiev, Khristo N.</creator><creator>Aliev, Ilham A.</creator><general>Springer US</general><general>Springer Nature B.V</general><scope>AAYXX</scope><scope>CITATION</scope></search><sort><creationdate>2020</creationdate><title>On values of the Riemann zeta function at positive integers</title><author>Dil, Ayhan ; Boyadzhiev, Khristo N. ; Aliev, Ilham A.</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c319t-f749af8e694f37077d200fd39d1a1f1f16e4bbf0adcd2481b279df0ca2b515f33</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2020</creationdate><topic>Actuarial Sciences</topic><topic>Integers</topic><topic>Mathematics</topic><topic>Mathematics and Statistics</topic><topic>Number Theory</topic><topic>Numbers</topic><topic>Ordinary Differential Equations</topic><topic>Probability Theory and Stochastic Processes</topic><topic>Sequences</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Dil, Ayhan</creatorcontrib><creatorcontrib>Boyadzhiev, Khristo N.</creatorcontrib><creatorcontrib>Aliev, Ilham A.</creatorcontrib><collection>CrossRef</collection><jtitle>Lithuanian mathematical journal</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Dil, Ayhan</au><au>Boyadzhiev, Khristo N.</au><au>Aliev, Ilham A.</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>On values of the Riemann zeta function at positive integers</atitle><jtitle>Lithuanian mathematical journal</jtitle><stitle>Lith Math J</stitle><date>2020</date><risdate>2020</risdate><volume>60</volume><issue>1</issue><spage>9</spage><epage>24</epage><pages>9-24</pages><issn>0363-1672</issn><eissn>1573-8825</eissn><abstract>We give new proofs of some known results on the values of the Riemann zeta function at positive integers and obtain some new theorems related to these values. Considering even zeta values as ζ(2
n
) =
η
n
π
2
n
, we obtain the generating functions of the sequences
η
n
and (−1)
n
η
n
. Using the Riemann–Lebesgue lemma, we give recurrence relations for ζ(2
n
) and ζ(2
n
+ 1). Furthermore, we prove some series equations for
∑
k
=
1
∞
−
1
k
−
1
ζ
p
+
k
/
k
.</abstract><cop>New York</cop><pub>Springer US</pub><doi>10.1007/s10986-019-09456-7</doi><tpages>16</tpages></addata></record> |
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| ispartof | Lithuanian mathematical journal, 2020, Vol.60 (1), p.9-24 |
| issn | 0363-1672 1573-8825 |
| language | eng |
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| source | Springer Nature - Complete Springer Journals |
| subjects | Actuarial Sciences Integers Mathematics Mathematics and Statistics Number Theory Numbers Ordinary Differential Equations Probability Theory and Stochastic Processes Sequences |
| title | On values of the Riemann zeta function at positive integers |
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