A note on S(t) and the zeros of the Riemann zeta-function
Let π S(t) denote the argument of the Riemann zeta-function at the point 1/2 + it. Assuming the Riemann hypothesis, we sharpen the constant in the best currently known bounds for S(t) and for the change of S(t) in intervals. We then deduce estimates for the largest multiplicity of a zero of the zeta...
Saved in:
| Published in: | The Bulletin of the London Mathematical Society Vol. 39; no. 3; pp. 482 - 486 |
|---|---|
| Main Authors: | , |
| Format: | Journal Article |
| Language: | English |
| Published: |
Oxford University Press
01.06.2007
|
| ISSN: | 0024-6093, 1469-2120 |
| Online Access: | Get full text |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| Abstract | Let π S(t) denote the argument of the Riemann zeta-function at the point 1/2 + it. Assuming the Riemann hypothesis, we sharpen the constant in the best currently known bounds for S(t) and for the change of S(t) in intervals. We then deduce estimates for the largest multiplicity of a zero of the zeta-function, and for the largest gap between the zeros. |
|---|---|
| AbstractList | Let π S(t) denote the argument of the Riemann zeta-function at the point 1/2 + it. Assuming the Riemann hypothesis, we sharpen the constant in the best currently known bounds for S(t) and for the change of S(t) in intervals. We then deduce estimates for the largest multiplicity of a zero of the zeta-function, and for the largest gap between the zeros. |
| Author | Gonek, S. M. Goldston, D. A. |
| Author_xml | – sequence: 1 givenname: D. A. surname: Goldston fullname: Goldston, D. A. email: goldston@math.sjsu.edu organization: Department of MathematicsSan Jose State UniversitySan Jose, CA 95192USAgoldston@math.sjsu.edu – sequence: 2 givenname: S. M. surname: Gonek fullname: Gonek, S. M. email: goldston@math.sjsu.edu organization: Department of MathematicsUniversity of RochesterRochester, NY 14627USAgonek@math.rochester.edu |
| BookMark | eNo9j81KAzEYRYMo2FZXvkCWuoj98tNkZlmrtUpFsArFTcjPNzjaycjMiNant7Xi6nIvnAunT_ZTnZCQEw7nnHMx9KuqHfpYgRR7pMeVzpngAvZJD0AopiGXh6Tftq8AXILhPZKPaao7pHWii9PujLoUafeC9BubuqV18VseSqxcSpuxc6z4SKEr63REDgq3avH4LwfkaXr1OJmx-f31zWQ8Z6WQI8FU5jA6pwK4TPvgeWYKFYwQceTByaBjwCgCD9qgctEDehmVMbnkQiuMckD47vezXOHavjdl5Zq15WC3ynarbHfK9mJ-twCViQ3DdkzZdvj1z7jmzWojzcjOls9WTS-z5a0SdiF_APEAXUk |
| ContentType | Journal Article |
| Copyright | 2007 London Mathematical Society |
| Copyright_xml | – notice: 2007 London Mathematical Society |
| DBID | BSCLL |
| DOI | 10.1112/blms/bdm032 |
| DatabaseName | Istex |
| DatabaseTitleList | |
| DeliveryMethod | fulltext_linktorsrc |
| Discipline | Mathematics |
| EISSN | 1469-2120 |
| EndPage | 486 |
| ExternalDocumentID | BLMS0482 ark_67375_HXZ_4FD8XJ42_S |
| Genre | article |
| GroupedDBID | --Z -~X .2P .I3 0R~ 1OB 1OC 1TH 23N 33P 4.4 5GY 5VS 6TJ 70D AAGQS AAHQN AAIJN AAJKP AAMMB AAMNL AAMVS AANLZ AAOGV AAPXW AASGY AASVR AAUQX AAVAP AAXRX AAYCA AAZKR ABCQX ABCUV ABEFU ABEJV ABEUO ABFSI ABGDZ ABGNP ABITZ ABIXL ABJNI ABLJU ABNGD ABNKS ABPTD ABQLI ABSMQ ABVKB ABXVV ABZBJ ACAHQ ACCZN ACFRR ACGFS ACIPB ACNCT ACPOU ACQPF ACUKT ACUTJ ACVCV ACXBN ACXQS ADBBV ADEOM ADEYI ADHKW ADHZD ADKYN ADMGS ADOCK ADOZA ADXAS ADZMN ADZXQ AECKG AEFGJ AEGPL AEIGN AEJOX AENEX AEPUE AETEA AEUYR AEYWJ AFBPY AFFPM AFGKR AFIYH AFKSM AFWVQ AFZJQ AGHNM AGKEF AGMDO AGQPQ AGSYK AGXDD AGYGG AHBTC AHXPO AI. AIDQK AIDYY AIJHB AIQQE AITYG AIURR AJEUX ALMA_UNASSIGNED_HOLDINGS ALTZX ALUQC ALUQN ALVPJ AMVHM AMYDB APJGH ASAOO ASPBG ATDFG AUFTA AVWKF AXUDD AZFZN BFHJK BMNLL BMXJE BQUQU BSCLL CAG CHEAL COF CS3 CXTWN CZ4 DCZOG DFGAJ DILTD DRFUL DRSTM DU5 D~K E.L EBS EE~ EJD F9B FEDTE H13 H5~ HAR HGLYW HVGLF HW0 H~9 IOX KOP KSI L7B L98 LATKE LEEKS LOXES LUTES LYRES M-Z MBTAY MEWTI MRFUL MRSTM MSFUL MSSTM MXFUL MXSTM N9A NGC O9- OHT P2P P2W PALCI PB- Q1. Q5Y RCA RD5 RJQFR ROL ROZ RW1 RXO SUPJJ TJP TN5 UPT VH1 WIH WIK WOHZO WXSBR X7H XKC YYP ZY4 ZZTAW ~91 |
| ID | FETCH-LOGICAL-i2352-48aedaa4c0a86bcb187f4c722d5b0a3c6dced2c1c67e4adb0eb3d477931264ed3 |
| IEDL.DBID | DRFUL |
| ISICitedReferencesCount | 41 |
| ISICitedReferencesURI | http://www.webofscience.com/api/gateway?GWVersion=2&SrcApp=Summon&SrcAuth=ProQuest&DestLinkType=CitingArticles&DestApp=WOS_CPL&KeyUT=000247860400018&url=https%3A%2F%2Fcvtisr.summon.serialssolutions.com%2F%23%21%2Fsearch%3Fho%3Df%26include.ft.matches%3Dt%26l%3Dnull%26q%3D |
| ISSN | 0024-6093 |
| IngestDate | Wed Aug 20 07:27:51 EDT 2025 Sat Sep 20 11:01:50 EDT 2025 |
| IsPeerReviewed | true |
| IsScholarly | true |
| Issue | 3 |
| Language | English |
| LinkModel | DirectLink |
| MergedId | FETCHMERGED-LOGICAL-i2352-48aedaa4c0a86bcb187f4c722d5b0a3c6dced2c1c67e4adb0eb3d477931264ed3 |
| Notes | ArticleID:bdm032 2000 Mathematics Subject Classification 11M26. ark:/67375/HXZ-4FD8XJ42-S istex:B3F619E9F5DAA0F420BCF623D15C2E223871CA47 2000 Mathematics Subject Classification 11M26. The research of both authors was supported in part by a National Science Foundation FRG grant (DMS 0244660). The first author was also partially supported by NSF grant DMS 0300563, and the second author by NSF grant DMS 0201457. The authors wish to thank the Isaac Newton Institute for its hospitality during their work on this paper, and also the American Institute of Mathematics. |
| PageCount | 5 |
| ParticipantIDs | wiley_primary_10_1112_blms_bdm032_BLMS0482 istex_primary_ark_67375_HXZ_4FD8XJ42_S |
| PublicationCentury | 2000 |
| PublicationDate | June 2007 |
| PublicationDateYYYYMMDD | 2007-06-01 |
| PublicationDate_xml | – month: 06 year: 2007 text: June 2007 |
| PublicationDecade | 2000 |
| PublicationTitle | The Bulletin of the London Mathematical Society |
| PublicationYear | 2007 |
| Publisher | Oxford University Press |
| Publisher_xml | – name: Oxford University Press |
| SSID | ssj0013071 |
| Score | 1.9586715 |
| Snippet | Let π S(t) denote the argument of the Riemann zeta-function at the point 1/2 + it. Assuming the Riemann hypothesis, we sharpen the constant in the best... Let π S(t) denote the argument of the Riemann zeta‐function at the point 1/2 + it. Assuming the Riemann hypothesis, we sharpen the constant in the best... |
| SourceID | wiley istex |
| SourceType | Publisher |
| StartPage | 482 |
| Title | A note on S(t) and the zeros of the Riemann zeta-function |
| URI | https://api.istex.fr/ark:/67375/HXZ-4FD8XJ42-S/fulltext.pdf https://onlinelibrary.wiley.com/doi/abs/10.1112%2Fblms%2Fbdm032 |
| Volume | 39 |
| WOSCitedRecordID | wos000247860400018&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: 1469-2120 dateEnd: 99991231 omitProxy: false ssIdentifier: ssj0013071 issn: 0024-6093 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/eLvHCXMwpV3LSsNAFB2kdaEL32J9MQsRFUIzk0kmxVW1hiJtkdZCcDPMK1DURNoo4spP8Bv9EmeSWHEpLpPcDOHkPs6dx70AHJmYyaWvtRMqLBySqMDhSHiOr8NAk6Tl8pYomk3QwSCM49ZN1efUnoUp60PMJ9ysZRT-2ho4F1UXEmSLhooHOxcRCfXoesYH17HRXb8G6p1hNO79LCS4tGyah4kTmOy9OqJnhmjaAZrl64aZWlBffzPUIsREq__-uDWwUrFL2C7VYR0s6HQDLPfnpVlnm-C8DdMs1zBL4egkP4U8VdA8hm_axEuYJcXFcGLk09TczPnn-4cNfvYHboFxdHV72XWqDgrOBBtm5ZCQa8U5kS4PAyEFCmlCJMVY-cLlngyU1ApLJAOqCVfCNam1ItTYLDJESStvG9TSLNU7AFrXwLEMJRGCIFtTRtLE8yRShmIowhvguICQPZVVMhif3ttNY9Rn3fiOkagTxtcEs1EDnBXIzQXLNAQzixkrEWMXvf7I-Ba8-xfhPbD0vZvPRfuglk-f9QFYlC_5ZDY9rDTjC4J5v-g |
| linkProvider | Wiley-Blackwell |
| linkToHtml | http://cvtisr.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwpV1LS8NAEF6kFdSDb7E-9yCiQmiy2SYpnqo1VG2L9AHFy7KvQFFTaaOIJ3-Cv9Ff4mwSqx7FYzaTZZnM45vZ3RmEDsBnclnR2goUERaNlGdxR7hWRQeeplHV5lWRNpvw2-1gMKje_LjFn9WHmCbcjGak9toouElI51puqoaKe5OMCIV6sF0wwkUKogQyXqx3wn7zeyfB9rOueYRaHoTv-R09mKJsJihnnwM0NVx9-Q1RUx8TLv1_dctoMceXuJYJxAqa0fEqWmhNi7NO1tBpDcejRONRjLtHyTHmscLwGr9q8Jh4FKUPnSHQxzEMJvzj7d24P_ML11E_vOidN6y8h4I1JICtLBpwrTin0uaBJ6RwAj-i0idEVYTNXekpqRWRjvR8TbkSNgTXivqgtQ5AJa3cDVSIR7HeRNgYB05kIKkQ1DFVZaQfua50FIAMRXkJHaY8ZI9ZnQzGx3fm2JhfYY3BLaNhPRhcUcK6JXSSsm5KmAUihBmesYxj7KzZ6oJ1IVt_Id5Hc41eq8mal-3rbTT_dbbPdnZQIRk_6V00K5-T4WS8l4vJJwj2w9g |
| linkToPdf | http://cvtisr.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwpV1LS8NAEF6kFdGDb7E-9yCiQmiy2WRTPFVr8NGW0looXpZ9BYqalDaKePIn-Bv9Je4msepRPGYzWZbJzsw3-_gGgAMdM5nwlLICibiFI-lbzOGu5anAVziq2azGs2ITpN0OBoNa58ct_pwfYrrgZiwj89fGwNVIRoWVG9ZQ_mAWI0IuH21XO-EyNoVkSqDc6Ib95vdOgk3yqnkIW75O34s7erqLqumgmn-uoanR6stviJrFmHDp_6NbBosFvoT1fEKsgBkVr4KF1pScdbIGTuswTlIFkxj2jtJjyGIJ9Wv4qnTEhEmUPXSHWj6OdWPKPt7eTfgzv3Ad9MOL2_NLq6ihYA2RxlYWDpiSjGFhs8DngjsBibAgCEmP28wVvhRKIuEInyjMJLd1ci0x0VbraKikpLsBSnESq00AjXNgSAQCc44dwyojSOS6wpEaZEjMKuAw0yEd5TwZlI3vzbEx4tHLwR3FYSMYXGNEexVwkqluKpgnIogandFcY_Ss2epp74K2_iK8D-Y6jZA2r9o322D-62if7eyAUjp-UrtgVjynw8l4r5gln-1dw1M |
| 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=A+note+on+S%28t%29+and+the+zeros+of+the+Riemann+zeta-function&rft.jtitle=The+Bulletin+of+the+London+Mathematical+Society&rft.au=Goldston%2C+D.+A.&rft.au=Gonek%2C+S.+M.&rft.date=2007-06-01&rft.pub=Oxford+University+Press&rft.issn=0024-6093&rft.eissn=1469-2120&rft.volume=39&rft.issue=3&rft.spage=482&rft.epage=486&rft_id=info:doi/10.1112%2Fblms%2Fbdm032&rft.externalDBID=n%2Fa&rft.externalDocID=ark_67375_HXZ_4FD8XJ42_S |
| thumbnail_l | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/lc.gif&issn=0024-6093&client=summon |
| thumbnail_m | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/mc.gif&issn=0024-6093&client=summon |
| thumbnail_s | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/sc.gif&issn=0024-6093&client=summon |