An improved bound on the least common multiple of polynomial sequences
Cilleruelo a conjecturé que si f ϵ ℤ[x] de degré d ≥ 2 est irréductible sur les rationnels, alors log lcm(f(1), . . . , f(N)) ∼ (d − 1)N logN quand N → ∞. Il l’a prouvé dans le cas d = 2. Très récemment, Maynard et Rudnick ont prouvé qu’il existe cd > 0 tel que log lcm((f(1), . . . , f(N)) ≳ c,d...
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| Veröffentlicht in: | Journal de theorie des nombres de bordeaux 2020-01, Vol.32 (3), p.891-899 |
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| description | Cilleruelo a conjecturé que si f ϵ ℤ[x] de degré d ≥ 2 est irréductible sur les rationnels, alors log lcm(f(1), . . . , f(N)) ∼ (d − 1)N logN quand N → ∞. Il l’a prouvé dans le cas d = 2. Très récemment, Maynard et Rudnick ont prouvé qu’il existe cd
> 0 tel que log lcm((f(1), . . . , f(N)) ≳ c,d
N logN, et ont montré qu’on peut prendre
c
d
=
d
−
1
d
2
. Nous donnons une preuve alternative de ce résultat avec la constante améliorée cd
= 1. De plus, nous prouvons la minoration log rad lcm
(
f
(
1
)
,
…
,
f
(
N
)
)
≥
2
d
N
log
N
et proposons une conjecture plus forte affirmant que log rad lcm(f(1), . . . , f(N)) ∼ (d − 1)N logN quand N → ∞.
Cilleruelo conjectured that if f ϵ ℤ[x] of degree d ≥ 2 is irreducible over the rationals, then log lcm(f(1), . . . , f(N)) ∼ (d − 1)N logN as N → ∞. He proved it for the case d = 2. Very recently, Maynard and Rudnick proved there exists cd
> 0 with log lcm(f(1), . . . , f(N)) ≳ cd
N logN, and showed one can take
c
d
=
d
−
1
d
2
. We give an alternative proof of this result with the improved constant cd
= 1. We additionally prove the bound log rad lcm
(
f
(
1
)
,
…
,
f
(
N
)
)
≥
2
d
N
log
N
and make the stronger conjecture that log rad lcm(f(1), . . . , f(N)) ∼ (d − 1)N logN as N → ∞. |
| doi_str_mv | 10.5802/jtnb.1146 |
| format | Article |
| fullrecord | <record><control><sourceid>jstor_webof</sourceid><recordid>TN_cdi_webofscience_primary_000609361000010CitationCount</recordid><sourceformat>XML</sourceformat><sourcesystem>PC</sourcesystem><jstor_id>26974697</jstor_id><sourcerecordid>26974697</sourcerecordid><originalsourceid>FETCH-LOGICAL-c251t-e5afa30ed397e263ee41a0226be02529294ac1d9aa3862a52205fd8729c3555a3</originalsourceid><addsrcrecordid>eNqNkMFKxDAQhoMouK4efAAhV5Guk0mTNseluCoseNFzSdspdmmb2mSVfXtbVvbsYZhh-P5h-Bi7FbBSKeDjLvTFSohYn7EFCpFGqUrwnC0ExjpKYlCX7Mr7HQBKbdIF26x73nTD6L6p4oXb9xV3PQ-fxFuyPvDSdd206PZtaIaWuKv54NpD77rGttzT1576kvw1u6ht6-nmry_Zx-bpPXuJtm_Pr9l6G5WoRIhI2dpKoEqahFBLolhYQNQFASo0aGJbispYK1ONViGCqqs0QVNKpZSVS3Z_vFuOzvuR6nwYm86Oh1xAPgvIZwH5LGBi0yP7Q4WrfdnMn554ANBgpBbTAAKyJtjQuD6bFIQp-vD_6ETfHemdD248YahNEk8lfwHShHjW</addsrcrecordid><sourcetype>Aggregation Database</sourcetype><iscdi>true</iscdi><recordtype>article</recordtype></control><display><type>article</type><title>An improved bound on the least common multiple of polynomial sequences</title><source>Elektronische Zeitschriftenbibliothek - Frei zugängliche E-Journals</source><source>JSTOR Mathematics & Statistics</source><source>JSTOR Archive Collection A-Z Listing</source><source>Web of Science - Science Citation Index Expanded - 2020<img src="https://exlibris-pub.s3.amazonaws.com/fromwos-v2.jpg" /></source><creator>SAH, Ashwin</creator><creatorcontrib>SAH, Ashwin</creatorcontrib><description>Cilleruelo a conjecturé que si f ϵ ℤ[x] de degré d ≥ 2 est irréductible sur les rationnels, alors log lcm(f(1), . . . , f(N)) ∼ (d − 1)N logN quand N → ∞. Il l’a prouvé dans le cas d = 2. Très récemment, Maynard et Rudnick ont prouvé qu’il existe cd
> 0 tel que log lcm((f(1), . . . , f(N)) ≳ c,d
N logN, et ont montré qu’on peut prendre
c
d
=
d
−
1
d
2
. Nous donnons une preuve alternative de ce résultat avec la constante améliorée cd
= 1. De plus, nous prouvons la minoration log rad lcm
(
f
(
1
)
,
…
,
f
(
N
)
)
≥
2
d
N
log
N
et proposons une conjecture plus forte affirmant que log rad lcm(f(1), . . . , f(N)) ∼ (d − 1)N logN quand N → ∞.
Cilleruelo conjectured that if f ϵ ℤ[x] of degree d ≥ 2 is irreducible over the rationals, then log lcm(f(1), . . . , f(N)) ∼ (d − 1)N logN as N → ∞. He proved it for the case d = 2. Very recently, Maynard and Rudnick proved there exists cd
> 0 with log lcm(f(1), . . . , f(N)) ≳ cd
N logN, and showed one can take
c
d
=
d
−
1
d
2
. We give an alternative proof of this result with the improved constant cd
= 1. We additionally prove the bound log rad lcm
(
f
(
1
)
,
…
,
f
(
N
)
)
≥
2
d
N
log
N
and make the stronger conjecture that log rad lcm(f(1), . . . , f(N)) ∼ (d − 1)N logN as N → ∞.</description><identifier>ISSN: 1246-7405</identifier><identifier>ISSN: 2118-8572</identifier><identifier>EISSN: 2118-8572</identifier><identifier>DOI: 10.5802/jtnb.1146</identifier><language>eng</language><publisher>TALENCE: Société Arithmétique de Bordeaux</publisher><subject>Mathematics ; Physical Sciences ; Science & Technology</subject><ispartof>Journal de theorie des nombres de bordeaux, 2020-01, Vol.32 (3), p.891-899</ispartof><rights>Société Arithmétique de Bordeaux, 2020</rights><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>true</woscitedreferencessubscribed><woscitedreferencescount>2</woscitedreferencescount><woscitedreferencesoriginalsourcerecordid>wos000609361000010</woscitedreferencesoriginalsourcerecordid><citedby>FETCH-LOGICAL-c251t-e5afa30ed397e263ee41a0226be02529294ac1d9aa3862a52205fd8729c3555a3</citedby><cites>FETCH-LOGICAL-c251t-e5afa30ed397e263ee41a0226be02529294ac1d9aa3862a52205fd8729c3555a3</cites></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktopdf>$$Uhttps://www.jstor.org/stable/pdf/26974697$$EPDF$$P50$$Gjstor$$H</linktopdf><linktohtml>$$Uhttps://www.jstor.org/stable/26974697$$EHTML$$P50$$Gjstor$$H</linktohtml><link.rule.ids>315,782,786,805,834,27931,27932,28255,58024,58028,58257,58261</link.rule.ids></links><search><creatorcontrib>SAH, Ashwin</creatorcontrib><title>An improved bound on the least common multiple of polynomial sequences</title><title>Journal de theorie des nombres de bordeaux</title><addtitle>J THEOR NOMBR BORDX</addtitle><description>Cilleruelo a conjecturé que si f ϵ ℤ[x] de degré d ≥ 2 est irréductible sur les rationnels, alors log lcm(f(1), . . . , f(N)) ∼ (d − 1)N logN quand N → ∞. Il l’a prouvé dans le cas d = 2. Très récemment, Maynard et Rudnick ont prouvé qu’il existe cd
> 0 tel que log lcm((f(1), . . . , f(N)) ≳ c,d
N logN, et ont montré qu’on peut prendre
c
d
=
d
−
1
d
2
. Nous donnons une preuve alternative de ce résultat avec la constante améliorée cd
= 1. De plus, nous prouvons la minoration log rad lcm
(
f
(
1
)
,
…
,
f
(
N
)
)
≥
2
d
N
log
N
et proposons une conjecture plus forte affirmant que log rad lcm(f(1), . . . , f(N)) ∼ (d − 1)N logN quand N → ∞.
Cilleruelo conjectured that if f ϵ ℤ[x] of degree d ≥ 2 is irreducible over the rationals, then log lcm(f(1), . . . , f(N)) ∼ (d − 1)N logN as N → ∞. He proved it for the case d = 2. Very recently, Maynard and Rudnick proved there exists cd
> 0 with log lcm(f(1), . . . , f(N)) ≳ cd
N logN, and showed one can take
c
d
=
d
−
1
d
2
. We give an alternative proof of this result with the improved constant cd
= 1. We additionally prove the bound log rad lcm
(
f
(
1
)
,
…
,
f
(
N
)
)
≥
2
d
N
log
N
and make the stronger conjecture that log rad lcm(f(1), . . . , f(N)) ∼ (d − 1)N logN as N → ∞.</description><subject>Mathematics</subject><subject>Physical Sciences</subject><subject>Science & Technology</subject><issn>1246-7405</issn><issn>2118-8572</issn><issn>2118-8572</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2020</creationdate><recordtype>article</recordtype><sourceid>AOWDO</sourceid><recordid>eNqNkMFKxDAQhoMouK4efAAhV5Guk0mTNseluCoseNFzSdspdmmb2mSVfXtbVvbsYZhh-P5h-Bi7FbBSKeDjLvTFSohYn7EFCpFGqUrwnC0ExjpKYlCX7Mr7HQBKbdIF26x73nTD6L6p4oXb9xV3PQ-fxFuyPvDSdd206PZtaIaWuKv54NpD77rGttzT1576kvw1u6ht6-nmry_Zx-bpPXuJtm_Pr9l6G5WoRIhI2dpKoEqahFBLolhYQNQFASo0aGJbispYK1ONViGCqqs0QVNKpZSVS3Z_vFuOzvuR6nwYm86Oh1xAPgvIZwH5LGBi0yP7Q4WrfdnMn554ANBgpBbTAAKyJtjQuD6bFIQp-vD_6ETfHemdD248YahNEk8lfwHShHjW</recordid><startdate>20200101</startdate><enddate>20200101</enddate><creator>SAH, Ashwin</creator><general>Société Arithmétique de Bordeaux</general><general>Univ Bordeaux, Inst Mathematiques Bordeaux</general><scope>AOWDO</scope><scope>BLEPL</scope><scope>DTL</scope><scope>AAYXX</scope><scope>CITATION</scope></search><sort><creationdate>20200101</creationdate><title>An improved bound on the least common multiple of polynomial sequences</title><author>SAH, Ashwin</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c251t-e5afa30ed397e263ee41a0226be02529294ac1d9aa3862a52205fd8729c3555a3</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2020</creationdate><topic>Mathematics</topic><topic>Physical Sciences</topic><topic>Science & Technology</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>SAH, Ashwin</creatorcontrib><collection>Web of Science - Science Citation Index Expanded - 2020</collection><collection>Web of Science Core Collection</collection><collection>Science Citation Index Expanded</collection><collection>CrossRef</collection><jtitle>Journal de theorie des nombres de bordeaux</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>SAH, Ashwin</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>An improved bound on the least common multiple of polynomial sequences</atitle><jtitle>Journal de theorie des nombres de bordeaux</jtitle><stitle>J THEOR NOMBR BORDX</stitle><date>2020-01-01</date><risdate>2020</risdate><volume>32</volume><issue>3</issue><spage>891</spage><epage>899</epage><pages>891-899</pages><issn>1246-7405</issn><issn>2118-8572</issn><eissn>2118-8572</eissn><abstract>Cilleruelo a conjecturé que si f ϵ ℤ[x] de degré d ≥ 2 est irréductible sur les rationnels, alors log lcm(f(1), . . . , f(N)) ∼ (d − 1)N logN quand N → ∞. Il l’a prouvé dans le cas d = 2. Très récemment, Maynard et Rudnick ont prouvé qu’il existe cd
> 0 tel que log lcm((f(1), . . . , f(N)) ≳ c,d
N logN, et ont montré qu’on peut prendre
c
d
=
d
−
1
d
2
. Nous donnons une preuve alternative de ce résultat avec la constante améliorée cd
= 1. De plus, nous prouvons la minoration log rad lcm
(
f
(
1
)
,
…
,
f
(
N
)
)
≥
2
d
N
log
N
et proposons une conjecture plus forte affirmant que log rad lcm(f(1), . . . , f(N)) ∼ (d − 1)N logN quand N → ∞.
Cilleruelo conjectured that if f ϵ ℤ[x] of degree d ≥ 2 is irreducible over the rationals, then log lcm(f(1), . . . , f(N)) ∼ (d − 1)N logN as N → ∞. He proved it for the case d = 2. Very recently, Maynard and Rudnick proved there exists cd
> 0 with log lcm(f(1), . . . , f(N)) ≳ cd
N logN, and showed one can take
c
d
=
d
−
1
d
2
. We give an alternative proof of this result with the improved constant cd
= 1. We additionally prove the bound log rad lcm
(
f
(
1
)
,
…
,
f
(
N
)
)
≥
2
d
N
log
N
and make the stronger conjecture that log rad lcm(f(1), . . . , f(N)) ∼ (d − 1)N logN as N → ∞.</abstract><cop>TALENCE</cop><pub>Société Arithmétique de Bordeaux</pub><doi>10.5802/jtnb.1146</doi><tpages>9</tpages></addata></record> |
| fulltext | fulltext |
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| language | eng |
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| source | Elektronische Zeitschriftenbibliothek - Frei zugängliche E-Journals; JSTOR Mathematics & Statistics; JSTOR Archive Collection A-Z Listing; Web of Science - Science Citation Index Expanded - 2020<img src="https://exlibris-pub.s3.amazonaws.com/fromwos-v2.jpg" /> |
| subjects | Mathematics Physical Sciences Science & Technology |
| title | An improved bound on the least common multiple of polynomial sequences |
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