Higher correlations of divisor sums related to primes II: variations of the error term in the prime number theorem
We calculate the triple correlations for the truncated divisor sum λR(n). The λR(n) behave over certain averages just as the prime counting von Mangoldt function Λ(n) does or is conjectured to do. We also calculate the mixed (with a factor of Λ(n)) correlations. The results for the moments up to the...
Gespeichert in:
| Veröffentlicht in: | Proceedings of the London Mathematical Society Jg. 95; H. 1; S. 199 - 247 |
|---|---|
| Hauptverfasser: | , |
| Format: | Journal Article |
| Sprache: | Englisch |
| Veröffentlicht: |
Oxford University Press
01.07.2007
|
| ISSN: | 0024-6115, 1460-244X |
| Online-Zugang: | Volltext |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| Abstract | We calculate the triple correlations for the truncated divisor sum λR(n). The λR(n) behave over certain averages just as the prime counting von Mangoldt function Λ(n) does or is conjectured to do. We also calculate the mixed (with a factor of Λ(n)) correlations. The results for the moments up to the third degree, and therefore the implications for the distribution of primes in short intervals, are the same as those we obtained (in the first paper with this title) by using the simpler approximation ΛR(n). However, when λR(n) is used, the error in the singular series approximation is often much smaller than what ΛR(n) allows. Assuming the Generalized Riemann Hypothesis (GRH) for Dirichlet L-functions, we obtain an Ω±-result for the variation of the error term in the prime number theorem. Formerly, our knowledge under GRH was restricted to Ω-results for the absolute value of this variation. An important ingredient in the last part of this work is a recent result due to Montgomery and Soundararajan which makes it possible for us to dispense with a large error term in the evaluation of a certain singular series average. We believe that our results on the sums λR(n) and ΛR(n) can be employed in diverse problems concerning primes. |
|---|---|
| AbstractList | We calculate the triple correlations for the truncated divisor sum λR(n). The λR(n) behave over certain averages just as the prime counting von Mangoldt function Λ(n) does or is conjectured to do. We also calculate the mixed (with a factor of Λ(n)) correlations. The results for the moments up to the third degree, and therefore the implications for the distribution of primes in short intervals, are the same as those we obtained (in the first paper with this title) by using the simpler approximation ΛR(n). However, when λR(n) is used, the error in the singular series approximation is often much smaller than what ΛR(n) allows. Assuming the Generalized Riemann Hypothesis (GRH) for DirichletL‐functions, we obtain an Ω±‐result for the variation of the error term in the prime number theorem. Formerly, our knowledge under GRH was restricted to Ω‐results for the absolute value of this variation. An important ingredient in the last part of this work is a recent result due to Montgomery and Soundararajan which makes it possible for us to dispense with a large error term in the evaluation of a certain singular series average. We believe that our results on the sums λR(n) and ΛR(n) can be employed in diverse problems concerning primes. We calculate the triple correlations for the truncated divisor sum λR(n). The λR(n) behave over certain averages just as the prime counting von Mangoldt function Λ(n) does or is conjectured to do. We also calculate the mixed (with a factor of Λ(n)) correlations. The results for the moments up to the third degree, and therefore the implications for the distribution of primes in short intervals, are the same as those we obtained (in the first paper with this title) by using the simpler approximation ΛR(n). However, when λR(n) is used, the error in the singular series approximation is often much smaller than what ΛR(n) allows. Assuming the Generalized Riemann Hypothesis (GRH) for Dirichlet L-functions, we obtain an Ω±-result for the variation of the error term in the prime number theorem. Formerly, our knowledge under GRH was restricted to Ω-results for the absolute value of this variation. An important ingredient in the last part of this work is a recent result due to Montgomery and Soundararajan which makes it possible for us to dispense with a large error term in the evaluation of a certain singular series average. We believe that our results on the sums λR(n) and ΛR(n) can be employed in diverse problems concerning primes. |
| Author | Yildirim, C. Y. Goldston, D. A. |
| Author_xml | – sequence: 1 givenname: D. A. surname: Goldston fullname: Goldston, D. A. email: goldston@math.sjsu.edu organization: San Jose State University – sequence: 2 givenname: C. Y. surname: Yildirim fullname: Yildirim, C. Y. email: yalciny@boun.edu.tr organization: Çengelköy |
| BookMark | eNp9kElPwzAQhS1UJNrCiT_gOwr1xLGbcIMKaFFZJAqquFhOMqGGLJWdFvrvSReBhASnkd68b5bXIa2yKpGQY2CnAOD35nnhevO0YMD2SBsCyTw_CKYt0mbMDzwJIA5Ix7k3xpjkXLSJHZrXGVqaVNZirmtTlY5WGU3N0rjKUrcoHN10MKV1RefWFOjoaHRGl9qaH6CeIUVrG6RGW1BTbpSNnZaLIm52NEJlsTgk-5nOHR7tapc8XV1OBkNvfH89GpyPvYQDj7xMZBxFqDOQkseogfNY6JiFHFIZ-FEqME4FC-MEtMikxCgLNYSRn8aBj6HPuwS2cxNbOWcxU4mpNwfXVptcAVPr0NQ6NLUNrWFOfjHrD7Rd_eHebfgwOa7-s6qH8e0jgyhqGG_LGFfj5zej7buSfd4Xajh9UeEELp5vBneK8S-HHZPl |
| CitedBy_id | crossref_primary_10_1007_s00208_021_02179_6 |
| ContentType | Journal Article |
| Copyright | 2007 London Mathematical Society |
| Copyright_xml | – notice: 2007 London Mathematical Society |
| DBID | BSCLL AAYXX CITATION |
| DOI | 10.1112/plms/pdm010 |
| DatabaseName | Istex CrossRef |
| DatabaseTitle | CrossRef |
| DatabaseTitleList | |
| DeliveryMethod | fulltext_linktorsrc |
| Discipline | Mathematics |
| EISSN | 1460-244X |
| EndPage | 247 |
| ExternalDocumentID | 10_1112_plms_pdm010 PLMS0199 ark_67375_HXZ_8T1BVJCN_0 |
| Genre | article |
| GroupedDBID | --Z -~X .2P .I3 0R~ 123 1OB 1OC 1TH 2WC 33P 4.4 5VS 6TJ 70D AAGQS AAHQN AAIJN AAJKP AAMMB AAMNL AAMVS AANLZ AAOGV AASGY AASVR AAUQX AAXRX AAYCA AAZKR ABCQX ABCUV ABEFU ABEJV ABEUO ABGDZ ABGNP ABITZ ABIXL ABJNI ABLJU ABNGD ABNKS ABQLI ABSMQ ABVKB ABXVV ABZBJ ACAHQ ACCZN ACGFS ACIPB ACNCT ACPOU ACQPF ACUKT ACXBN ACXQS ADBBV ADEOM ADEYI ADHZD ADKYN ADMGS ADOCK ADOZA ADXAS ADXHL ADZMN ADZXQ AECKG AEFGJ AEGPL AEIGN AEJOX AENEX AEPUE AETEA AEUYR AEYWJ AFBPY AFFNX AFFPM AFGKR AFIYH AFKSM AFWVQ AFZJQ AGHNM AGKEF AGQPQ AGSYK AGXDD AGYGG AHBTC AHXPO AIDQK AIDYY AIJHB AIQQE AITYG AIURR AJEUX ALMA_UNASSIGNED_HOLDINGS ALTZX ALUQC ALUQN ALVPJ AMVHM AMYDB ASAOO ASPBG ATDFG AUFTA AVWKF AXUDD AZFZN BFHJK BMNLL BMXJE BQUQU BSCLL CAG CHEAL COF CS3 CXTWN CZ4 D0L DCZOG DFGAJ DILTD DRFUL DRSTM DU5 D~K EBS EE~ EJD F9B FEDTE FSPIC H13 H5~ HAR HGLYW HVGLF HW0 H~9 IOX KOP L7B L98 LATKE LEEKS LH4 LOXES LPU LUTES LYRES M-Z MBTAY MEWTI MRFUL MRSTM MSFUL MSSTM MVM MXFUL MXSTM N9A NGC NU- O0~ O9- OHT O~Y P2P P2W PALCI PB- Q1. Q5Y RCA RD5 RJQFR ROL ROZ RW1 RXO S10 SAMSI SUPJJ TJP TN5 TUQ UQL WH7 WIH WIK WOHZO WXSBR X7H XJT XKC XOL XSW Y6R YNT YYP ZCG ZY4 ZZTAW ~91 AAHHS AAOIN ABDBF ABQTQ ABSAR ABTAH ACCFJ ACUFI ADRIX AEEZP AEQDE AFPWT AIWBW AJBDE ESX J21 KSI M49 ROX TCN AAYXX CITATION |
| ID | FETCH-LOGICAL-c3139-f5f3e58af1663bea133b5ab0831d6429d5ebd508bc1a5f66e9f8a1892db42e823 |
| IEDL.DBID | DRFUL |
| ISICitedReferencesCount | 1 |
| ISICitedReferencesURI | http://www.webofscience.com/api/gateway?GWVersion=2&SrcApp=Summon&SrcAuth=ProQuest&DestLinkType=CitingArticles&DestApp=WOS_CPL&KeyUT=000247956200007&url=https%3A%2F%2Fcvtisr.summon.serialssolutions.com%2F%23%21%2Fsearch%3Fho%3Df%26include.ft.matches%3Dt%26l%3Dnull%26q%3D |
| ISSN | 0024-6115 |
| IngestDate | Sat Nov 29 06:18:13 EST 2025 Tue Nov 18 22:36:48 EST 2025 Wed Jan 22 17:09:13 EST 2025 Sat Sep 20 11:01:45 EDT 2025 |
| IsPeerReviewed | true |
| IsScholarly | true |
| Issue | 1 |
| Language | English |
| License | http://doi.wiley.com/10.1002/tdm_license_1.1 http://onlinelibrary.wiley.com/termsAndConditions#vor |
| LinkModel | DirectLink |
| MergedId | FETCHMERGED-LOGICAL-c3139-f5f3e58af1663bea133b5ab0831d6429d5ebd508bc1a5f66e9f8a1892db42e823 |
| Notes | istex:B222091C16D15E745261E401C3F5747782C823B1 ark:/67375/HXZ-8T1BVJCN-0 2000 Mathematics Subject Classification 11N05 (primary), 11P32 (secondary). ArticleID:pdm010 2000 Mathematics Subject Classification The research of Goldston was supported by the NSF; that of Yildirim was supported by TÜBİTAK. 11N05 (primary), 11P32 (secondary). |
| PageCount | 49 |
| ParticipantIDs | crossref_citationtrail_10_1112_plms_pdm010 crossref_primary_10_1112_plms_pdm010 wiley_primary_10_1112_plms_pdm010_PLMS0199 istex_primary_ark_67375_HXZ_8T1BVJCN_0 |
| PublicationCentury | 2000 |
| PublicationDate | 2007-07 July 2007 2007-07-00 |
| PublicationDateYYYYMMDD | 2007-07-01 |
| PublicationDate_xml | – month: 07 year: 2007 text: 2007-07 |
| PublicationDecade | 2000 |
| PublicationTitle | Proceedings of the London Mathematical Society |
| PublicationYear | 2007 |
| Publisher | Oxford University Press |
| Publisher_xml | – name: Oxford University Press |
| SSID | ssj0006335 |
| Score | 1.7424363 |
| Snippet | We calculate the triple correlations for the truncated divisor sum λR(n). The λR(n) behave over certain averages just as the prime counting von Mangoldt... |
| SourceID | crossref wiley istex |
| SourceType | Enrichment Source Index Database Publisher |
| StartPage | 199 |
| Title | Higher correlations of divisor sums related to primes II: variations of the error term in the prime number theorem |
| URI | https://api.istex.fr/ark:/67375/HXZ-8T1BVJCN-0/fulltext.pdf https://onlinelibrary.wiley.com/doi/abs/10.1112%2Fplms%2Fpdm010 |
| Volume | 95 |
| WOSCitedRecordID | wos000247956200007&url=https%3A%2F%2Fcvtisr.summon.serialssolutions.com%2F%23%21%2Fsearch%3Fho%3Df%26include.ft.matches%3Dt%26l%3Dnull%26q%3D |
| hasFullText | 1 |
| inHoldings | 1 |
| isFullTextHit | |
| isPrint | |
| journalDatabaseRights | – providerCode: PRVWIB databaseName: Wiley Online Library Full Collection 2020 customDbUrl: eissn: 1460-244X dateEnd: 99991231 omitProxy: false ssIdentifier: ssj0006335 issn: 0024-6115 databaseCode: DRFUL dateStart: 19970101 isFulltext: true titleUrlDefault: https://onlinelibrary.wiley.com providerName: Wiley-Blackwell |
| link | http://cvtisr.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwpV1LSwMxEA7SetCDb7G-yKF4KCxtdjdt1puvUqUtRVspXpY8QbQPdqv4851s1pWCCOJ1MxmWzGTyTcjMh1A1DFUkAaZ6UpuGF_KAeIyz0KNGWrQchMb1me22-n02HkeDnOfU1sK4_hDFhZvdGVm8thuci5yFhNimofNXexfRnqtJVmJV9sF3aQmVr-_bo24RjJuB49iEowiyJELzEj1QUbcK6m760qFUtuv7sQxWs9Omvfnv_9xCGznQxBfOM7bRip7uoPVe0aU13UWJe-SBpWXoyN_E4ZnBtkQrnSUYnDTF2YhWeDHDc0sFkOLb23P8Din29wTQiXWSwBQb6PHzNPuSiWNHOYJdweRkD43aN8OrjpdzMHgyAHDoGWoCTRk3BAwnNIeUVlAuLD-ZgtQlUlQLBSBPSMKpaTZ1ZBgnLPKVCH3N_GAflaazqT5AmAeREiAjW4qFEEmECanxhWZEK6V4s4JqX0aIZd6g3PJkvMYuUfFju5SxW8gKqhbCc9eX42exs8yahQxPXuxTthaNO-OnmA3J5ePdVT8GwVpmxN-UxYNu7wEgcXT4F-EjtPZ1H9wgx6i0SN70CVqV74vnNDnN_fUTHlf1CQ |
| linkProvider | Wiley-Blackwell |
| linkToHtml | http://cvtisr.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwpV1LaxsxEBYhDqQ5NGnaEjeP6mB6MCzJ7kq2trfEibHbtTGpU0wvQk8I9Ytd1_Tnd7Rab2MogZKrdzQYzWj0zaCZD6EGITpRAFMDZexVQEQcBkwwElCrHFqOifVzZtP2cMgmk2T0pIvfz4eoCm7uZBTx2h1wV5AuT7mbGrqcumJEd6lnRY9VjYArgY_Xbu-7D2kVjVuxJ9mEuwjSpJCWPXqg4tIpuPTLt26lmtvg39totbhuuocv_6NH6HUJNfG19403aMfMj9HBoJrTmr9FmX_mgZXj6ChfxeGFxa5JK19kGNw0x8UXo_FqgZeODCDH_f5nvIYk--8C0IlNlsESF-rx47z4pRDHnnQE-5bJ2Tv00L0bd3pBycIQqBjgYWCpjQ1lwoZgOmkEJLWSCukYyjQkL4mmRmqAeVKFgtpWyySWiZAlkZYkMiyK36Pd-WJuThAWcaIlyKi2ZgRiibSE2kgaFhqttWjVUXNjBa7KEeWOKWPKfaoScbeV3G9kHTUq4aWfzPFvsU-FOSsZkf10j9nalPcmPzgbhzffv3SGHASbhRWfU8ZH6eAbgOLkw_8If0T7vfEg5Wl_-PUUvdpUh6_CM7S7yn6Zc7Sn1qvHPLsonfcPuGr4-Q |
| linkToPdf | http://cvtisr.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwpV1bSyMxFA7SirgP62UVXW95EB-EQWcmaTO-aXWwWkvxspR9CbmCqO0w0y3-fE8m46ggwuJr58uh5CQn3wk550NolxCdKKCpgTL2MCAiDgMmGAmoVY4tx8T6PrO9dr_PhsNk8K6K3_eHqC_c3M4o47Xb4CbTttrlrmto9uguI9JMP5U1Vk3ihGQaqHl6nd716mjcir3IJpxFkCaFtKrRAxMHzsCBH_7hVGq6CX7-yFbL4yZd-P4fXUQ_K6qJj_3aWEIzZrSMflzVfVqLXyj3zzywchod1as4PLbYFWkV4xzDMi1w-cVoPBnjzIkBFLjbPcJTSLLfBoBNbPIchrhQj-9H5S8lHHvREexLJp9W0F16dts5DyoVhkDFQA8DS21sKBM2BNdJIyCplVRIp1CmIXlJNDVSA82TKhTUtlomsUyELIm0JJFhUbyKGqPxyKwhLOJES8CotmYEYom0hNpIGhYarbVoraP9Vy9wVbUod0oZj9ynKhF3U8n9RK6j3Rqc-c4cn8P2SnfWGJE_uMdsbcrPh385uw1P_lx0-hyA-6UXvzLGB72rGyDFye__Ae-gucFpynvd_uUGmn-9HD4MN1Fjkv8zW2hWTSf3Rb5drd0XUJb4dA |
| openUrl | ctx_ver=Z39.88-2004&ctx_enc=info%3Aofi%2Fenc%3AUTF-8&rfr_id=info%3Asid%2Fsummon.serialssolutions.com&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Higher+correlations+of+divisor+sums+related+to+primes+II%3A+variations+of+the+error+term+in+the+prime+number+theorem&rft.jtitle=Proceedings+of+the+London+Mathematical+Society&rft.au=Goldston%2C+D.+A.&rft.au=Yildirim%2C+C.+Y.&rft.date=2007-07-01&rft.pub=Oxford+University+Press&rft.issn=0024-6115&rft.eissn=1460-244X&rft.volume=95&rft.issue=1&rft.spage=199&rft.epage=247&rft_id=info:doi/10.1112%2Fplms%2Fpdm010&rft.externalDBID=10.1112%252Fplms%252Fpdm010&rft.externalDocID=PLMS0199 |
| thumbnail_l | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/lc.gif&issn=0024-6115&client=summon |
| thumbnail_m | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/mc.gif&issn=0024-6115&client=summon |
| thumbnail_s | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/sc.gif&issn=0024-6115&client=summon |