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Evaluation of convolution sums ∑l+15m=nσ(l)σ(m) and ∑3l+5m=nσ(l)σ(m)
In this article, we have evaluated the convolution sums ∑ a l + b m = n σ ( l ) σ ( m ) for a · b = 15 , where a , b ∈ N , using an elementary method. The deduced convolution sums are in a little elegent form than that derived by B. Ramakrishnan and B. Sahu [ 24 ]. As a consequence, we determine a f...
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| Published in: | Indian journal of pure and applied mathematics 2022, Vol.53 (4), p.1110-1121 |
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| Main Authors: | , |
| Format: | Article |
| Language: | English |
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| cites | cdi_FETCH-LOGICAL-p142z-6eab30b02c86ccce9598811369fea47b2d4927585f95c79ea5253755ba4b386b3 |
| container_end_page | 1121 |
| container_issue | 4 |
| container_start_page | 1110 |
| container_title | Indian journal of pure and applied mathematics |
| container_volume | 53 |
| creator | Pushpa, K. Vasuki, K. R. |
| description | In this article, we have evaluated the convolution sums
∑
a
l
+
b
m
=
n
σ
(
l
)
σ
(
m
)
for
a
·
b
=
15
, where
a
,
b
∈
N
, using an elementary method. The deduced convolution sums are in a little elegent form than that derived by B. Ramakrishnan and B. Sahu [
24
]. As a consequence, we determine a formula for the number of representations of a positive integer
n
by the octonary quadratic form
x
1
2
+
x
1
x
2
+
x
2
2
+
x
3
2
+
x
3
x
4
+
x
4
2
+
5
x
5
2
+
x
5
x
6
+
x
6
2
+
x
7
2
+
x
7
x
8
+
x
8
2
. |
| doi_str_mv | 10.1007/s13226-022-00222-z |
| format | article |
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∑
a
l
+
b
m
=
n
σ
(
l
)
σ
(
m
)
for
a
·
b
=
15
, where
a
,
b
∈
N
, using an elementary method. The deduced convolution sums are in a little elegent form than that derived by B. Ramakrishnan and B. Sahu [
24
]. As a consequence, we determine a formula for the number of representations of a positive integer
n
by the octonary quadratic form
x
1
2
+
x
1
x
2
+
x
2
2
+
x
3
2
+
x
3
x
4
+
x
4
2
+
5
x
5
2
+
x
5
x
6
+
x
6
2
+
x
7
2
+
x
7
x
8
+
x
8
2
.</description><identifier>ISSN: 0019-5588</identifier><identifier>EISSN: 0975-7465</identifier><identifier>DOI: 10.1007/s13226-022-00222-z</identifier><language>eng</language><publisher>New Delhi: Indian National Science Academy</publisher><subject>Applications of Mathematics ; Convolution ; Mathematics ; Mathematics and Statistics ; Numerical Analysis ; Original Research ; Quadratic forms ; Sums</subject><ispartof>Indian journal of pure and applied mathematics, 2022, Vol.53 (4), p.1110-1121</ispartof><rights>The Indian National Science Academy 2022</rights><rights>The Indian National Science Academy 2022.</rights><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>false</woscitedreferencessubscribed><cites>FETCH-LOGICAL-p142z-6eab30b02c86ccce9598811369fea47b2d4927585f95c79ea5253755ba4b386b3</cites></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><link.rule.ids>314,777,781,27905,27906</link.rule.ids></links><search><creatorcontrib>Pushpa, K.</creatorcontrib><creatorcontrib>Vasuki, K. R.</creatorcontrib><title>Evaluation of convolution sums ∑l+15m=nσ(l)σ(m) and ∑3l+5m=nσ(l)σ(m)</title><title>Indian journal of pure and applied mathematics</title><addtitle>Indian J Pure Appl Math</addtitle><description>In this article, we have evaluated the convolution sums
∑
a
l
+
b
m
=
n
σ
(
l
)
σ
(
m
)
for
a
·
b
=
15
, where
a
,
b
∈
N
, using an elementary method. The deduced convolution sums are in a little elegent form than that derived by B. Ramakrishnan and B. Sahu [
24
]. As a consequence, we determine a formula for the number of representations of a positive integer
n
by the octonary quadratic form
x
1
2
+
x
1
x
2
+
x
2
2
+
x
3
2
+
x
3
x
4
+
x
4
2
+
5
x
5
2
+
x
5
x
6
+
x
6
2
+
x
7
2
+
x
7
x
8
+
x
8
2
.</description><subject>Applications of Mathematics</subject><subject>Convolution</subject><subject>Mathematics</subject><subject>Mathematics and Statistics</subject><subject>Numerical Analysis</subject><subject>Original Research</subject><subject>Quadratic forms</subject><subject>Sums</subject><issn>0019-5588</issn><issn>0975-7465</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2022</creationdate><recordtype>article</recordtype><sourceid/><recordid>eNpVkM1Kw0AUhQdRsFZfwFXAjaWM3pnJnZ-FCyn1BwJudD1M0om0pEnMNF1k68atT-Y7-CSmjSBuzr2Xc7gHPkLOGVwxAHUdmOBcUuCcQi-cdgdkBEYhVbHEw34HZiii1sfkJIQVgBRgzIgk860rWrdZVmVU5VFWlduqaPdnaNch-v74LKYM1zfl1_tlMellPYlcudgZopj-N07JUe6K4M9-55i83M2fZw80ebp_nN0mtGYx76j0LhWQAs-0zLLMGzRaMyakyb2LVcoXseEKNeYGM2W8Q45CIaYuToWWqRiTi-Fv3VRvrQ8bu6rapuwrLVeCKcm1UX1KDKlQN8vy1Td_KQZ2h80O2GxPzO6x2U78AOdKYZI</recordid><startdate>2022</startdate><enddate>2022</enddate><creator>Pushpa, K.</creator><creator>Vasuki, K. R.</creator><general>Indian National Science Academy</general><general>Springer Nature B.V</general><scope/></search><sort><creationdate>2022</creationdate><title>Evaluation of convolution sums ∑l+15m=nσ(l)σ(m) and ∑3l+5m=nσ(l)σ(m)</title><author>Pushpa, K. ; Vasuki, K. R.</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-p142z-6eab30b02c86ccce9598811369fea47b2d4927585f95c79ea5253755ba4b386b3</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2022</creationdate><topic>Applications of Mathematics</topic><topic>Convolution</topic><topic>Mathematics</topic><topic>Mathematics and Statistics</topic><topic>Numerical Analysis</topic><topic>Original Research</topic><topic>Quadratic forms</topic><topic>Sums</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Pushpa, K.</creatorcontrib><creatorcontrib>Vasuki, K. R.</creatorcontrib><jtitle>Indian journal of pure and applied mathematics</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Pushpa, K.</au><au>Vasuki, K. R.</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Evaluation of convolution sums ∑l+15m=nσ(l)σ(m) and ∑3l+5m=nσ(l)σ(m)</atitle><jtitle>Indian journal of pure and applied mathematics</jtitle><stitle>Indian J Pure Appl Math</stitle><date>2022</date><risdate>2022</risdate><volume>53</volume><issue>4</issue><spage>1110</spage><epage>1121</epage><pages>1110-1121</pages><issn>0019-5588</issn><eissn>0975-7465</eissn><abstract>In this article, we have evaluated the convolution sums
∑
a
l
+
b
m
=
n
σ
(
l
)
σ
(
m
)
for
a
·
b
=
15
, where
a
,
b
∈
N
, using an elementary method. The deduced convolution sums are in a little elegent form than that derived by B. Ramakrishnan and B. Sahu [
24
]. As a consequence, we determine a formula for the number of representations of a positive integer
n
by the octonary quadratic form
x
1
2
+
x
1
x
2
+
x
2
2
+
x
3
2
+
x
3
x
4
+
x
4
2
+
5
x
5
2
+
x
5
x
6
+
x
6
2
+
x
7
2
+
x
7
x
8
+
x
8
2
.</abstract><cop>New Delhi</cop><pub>Indian National Science Academy</pub><doi>10.1007/s13226-022-00222-z</doi><tpages>12</tpages></addata></record> |
| fulltext | fulltext |
| identifier | ISSN: 0019-5588 |
| ispartof | Indian journal of pure and applied mathematics, 2022, Vol.53 (4), p.1110-1121 |
| issn | 0019-5588 0975-7465 |
| language | eng |
| recordid | cdi_proquest_journals_2731762897 |
| source | Springer Nature |
| subjects | Applications of Mathematics Convolution Mathematics Mathematics and Statistics Numerical Analysis Original Research Quadratic forms Sums |
| title | Evaluation of convolution sums ∑l+15m=nσ(l)σ(m) and ∑3l+5m=nσ(l)σ(m) |
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