{"title":"作为算子的点谱的黎曼函数的零的集合","authors":"V. Kapustin","doi":"10.1090/spmj/1720","DOIUrl":null,"url":null,"abstract":"<p>A possible way of proving the Riemann hypothesis consists of constructing a selfadjoint operartor whose spectrum coincides with the set <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"left-brace z colon StartAbsoluteValue upper I m z EndAbsoluteValue greater-than one half comma zeta left-parenthesis one half minus i z right-parenthesis equals 0 right-brace\">\n <mml:semantics>\n <mml:mrow>\n <mml:mo fence=\"false\" stretchy=\"false\">{</mml:mo>\n <mml:mi>z</mml:mi>\n <mml:mspace width=\"thinmathspace\" />\n <mml:mo>:</mml:mo>\n <mml:mspace width=\"thinmathspace\" />\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mo stretchy=\"false\">|</mml:mo>\n </mml:mrow>\n <mml:mi>Im</mml:mi>\n <mml:mo><!-- --></mml:mo>\n <mml:mi>z</mml:mi>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mo stretchy=\"false\">|</mml:mo>\n </mml:mrow>\n <mml:mo>></mml:mo>\n <mml:mfrac>\n <mml:mn>1</mml:mn>\n <mml:mn>2</mml:mn>\n </mml:mfrac>\n <mml:mo>,</mml:mo>\n <mml:mtext> </mml:mtext>\n <mml:mi>ζ<!-- ζ --></mml:mi>\n <mml:mstyle scriptlevel=\"0\">\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mo maxsize=\"1.2em\" minsize=\"1.2em\">(</mml:mo>\n </mml:mrow>\n </mml:mstyle>\n <mml:mfrac>\n <mml:mn>1</mml:mn>\n <mml:mn>2</mml:mn>\n </mml:mfrac>\n <mml:mo>−<!-- − --></mml:mo>\n <mml:mi>i</mml:mi>\n <mml:mi>z</mml:mi>\n <mml:mstyle scriptlevel=\"0\">\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mo maxsize=\"1.2em\" minsize=\"1.2em\">)</mml:mo>\n </mml:mrow>\n </mml:mstyle>\n <mml:mo>=</mml:mo>\n <mml:mn>0</mml:mn>\n <mml:mo fence=\"false\" stretchy=\"false\">}</mml:mo>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">\\{z\\,: \\, |\\operatorname {Im}z|>\\frac 12, \\ \\zeta \\big (\\frac {1}{2}-iz\\big )=0\\}</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>. In the paper we construct a rank-one perturbation of a selfadjoint operator related to a certain canonical system for which a similar property is fulfilled.</p>","PeriodicalId":51162,"journal":{"name":"St Petersburg Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.7000,"publicationDate":"2022-06-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"The set of zeros of the Riemann zeta function as the point spectrum of an operator\",\"authors\":\"V. Kapustin\",\"doi\":\"10.1090/spmj/1720\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>A possible way of proving the Riemann hypothesis consists of constructing a selfadjoint operartor whose spectrum coincides with the set <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"left-brace z colon StartAbsoluteValue upper I m z EndAbsoluteValue greater-than one half comma zeta left-parenthesis one half minus i z right-parenthesis equals 0 right-brace\\\">\\n <mml:semantics>\\n <mml:mrow>\\n <mml:mo fence=\\\"false\\\" stretchy=\\\"false\\\">{</mml:mo>\\n <mml:mi>z</mml:mi>\\n <mml:mspace width=\\\"thinmathspace\\\" />\\n <mml:mo>:</mml:mo>\\n <mml:mspace width=\\\"thinmathspace\\\" />\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mo stretchy=\\\"false\\\">|</mml:mo>\\n </mml:mrow>\\n <mml:mi>Im</mml:mi>\\n <mml:mo><!-- --></mml:mo>\\n <mml:mi>z</mml:mi>\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mo stretchy=\\\"false\\\">|</mml:mo>\\n </mml:mrow>\\n <mml:mo>></mml:mo>\\n <mml:mfrac>\\n <mml:mn>1</mml:mn>\\n <mml:mn>2</mml:mn>\\n </mml:mfrac>\\n <mml:mo>,</mml:mo>\\n <mml:mtext> </mml:mtext>\\n <mml:mi>ζ<!-- ζ --></mml:mi>\\n <mml:mstyle scriptlevel=\\\"0\\\">\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mo maxsize=\\\"1.2em\\\" minsize=\\\"1.2em\\\">(</mml:mo>\\n </mml:mrow>\\n </mml:mstyle>\\n <mml:mfrac>\\n <mml:mn>1</mml:mn>\\n <mml:mn>2</mml:mn>\\n </mml:mfrac>\\n <mml:mo>−<!-- − --></mml:mo>\\n <mml:mi>i</mml:mi>\\n <mml:mi>z</mml:mi>\\n <mml:mstyle scriptlevel=\\\"0\\\">\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mo maxsize=\\\"1.2em\\\" minsize=\\\"1.2em\\\">)</mml:mo>\\n </mml:mrow>\\n </mml:mstyle>\\n <mml:mo>=</mml:mo>\\n <mml:mn>0</mml:mn>\\n <mml:mo fence=\\\"false\\\" stretchy=\\\"false\\\">}</mml:mo>\\n </mml:mrow>\\n <mml:annotation encoding=\\\"application/x-tex\\\">\\\\{z\\\\,: \\\\, |\\\\operatorname {Im}z|>\\\\frac 12, \\\\ \\\\zeta \\\\big (\\\\frac {1}{2}-iz\\\\big )=0\\\\}</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula>. In the paper we construct a rank-one perturbation of a selfadjoint operator related to a certain canonical system for which a similar property is fulfilled.</p>\",\"PeriodicalId\":51162,\"journal\":{\"name\":\"St Petersburg Mathematical Journal\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.7000,\"publicationDate\":\"2022-06-27\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"St Petersburg Mathematical Journal\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1090/spmj/1720\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"St Petersburg Mathematical Journal","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1090/spmj/1720","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
The set of zeros of the Riemann zeta function as the point spectrum of an operator
A possible way of proving the Riemann hypothesis consists of constructing a selfadjoint operartor whose spectrum coincides with the set {z:|Imz|>12,ζ(12−iz)=0}\{z\,: \, |\operatorname {Im}z|>\frac 12, \ \zeta \big (\frac {1}{2}-iz\big )=0\}. In the paper we construct a rank-one perturbation of a selfadjoint operator related to a certain canonical system for which a similar property is fulfilled.
期刊介绍：
This journal is a cover-to-cover translation into English of Algebra i Analiz, published six times a year by the mathematics section of the Russian Academy of Sciences.
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