{"title":"Bounds for the Tornheim double zeta function","authors":"Takashi Nakamura","doi":"10.1090/bproc/142","DOIUrl":null,"url":null,"abstract":"<p>In the present paper, we give bounds for the Tornheim double zeta function <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper T left-parenthesis s 1 comma s 2 comma s 3 right-parenthesis\">\n <mml:semantics>\n <mml:mrow>\n <mml:mi>T</mml:mi>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:msub>\n <mml:mi>s</mml:mi>\n <mml:mn>1</mml:mn>\n </mml:msub>\n <mml:mo>,</mml:mo>\n <mml:msub>\n <mml:mi>s</mml:mi>\n <mml:mn>2</mml:mn>\n </mml:msub>\n <mml:mo>,</mml:mo>\n <mml:msub>\n <mml:mi>s</mml:mi>\n <mml:mn>3</mml:mn>\n </mml:msub>\n <mml:mo stretchy=\"false\">)</mml:mo>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">T(s_1,s_2,s_3)</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> when <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"StartAbsoluteValue t 1 EndAbsoluteValue comma StartAbsoluteValue t 2 EndAbsoluteValue comma StartAbsoluteValue t 3 EndAbsoluteValue greater-than-or-equal-to 1\">\n <mml:semantics>\n <mml:mrow>\n <mml:mo fence=\"false\" stretchy=\"false\">|<!-- | --></mml:mo>\n <mml:msub>\n <mml:mi>t</mml:mi>\n <mml:mn>1</mml:mn>\n </mml:msub>\n <mml:mo fence=\"false\" stretchy=\"false\">|<!-- | --></mml:mo>\n <mml:mo>,</mml:mo>\n <mml:mo fence=\"false\" stretchy=\"false\">|<!-- | --></mml:mo>\n <mml:msub>\n <mml:mi>t</mml:mi>\n <mml:mn>2</mml:mn>\n </mml:msub>\n <mml:mo fence=\"false\" stretchy=\"false\">|<!-- | --></mml:mo>\n <mml:mo>,</mml:mo>\n <mml:mo fence=\"false\" stretchy=\"false\">|<!-- | --></mml:mo>\n <mml:msub>\n <mml:mi>t</mml:mi>\n <mml:mn>3</mml:mn>\n </mml:msub>\n <mml:mo fence=\"false\" stretchy=\"false\">|<!-- | --></mml:mo>\n <mml:mo>≥<!-- ≥ --></mml:mo>\n <mml:mn>1</mml:mn>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">\\lvert t_1 \\rvert , \\lvert t_2 \\rvert , \\lvert t_3 \\rvert \\ge 1</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>, <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"StartAbsoluteValue t 1 plus t 2 EndAbsoluteValue comma StartAbsoluteValue t 2 plus t 3 EndAbsoluteValue comma StartAbsoluteValue t 3 plus t 1 EndAbsoluteValue greater-than-or-equal-to 1\">\n <mml:semantics>\n <mml:mrow>\n <mml:mo fence=\"false\" stretchy=\"false\">|<!-- | --></mml:mo>\n <mml:msub>\n <mml:mi>t</mml:mi>\n <mml:mn>1</mml:mn>\n </mml:msub>\n <mml:mo>+</mml:mo>\n <mml:msub>\n <mml:mi>t</mml:mi>\n <mml:mn>2</mml:mn>\n </mml:msub>\n <mml:mo fence=\"false\" stretchy=\"false\">|<!-- | --></mml:mo>\n <mml:mo>,</mml:mo>\n <mml:mo fence=\"false\" stretchy=\"false\">|<!-- | --></mml:mo>\n <mml:msub>\n <mml:mi>t</mml:mi>\n <mml:mn>2</mml:mn>\n </mml:msub>\n <mml:mo>+</mml:mo>\n <mml:msub>\n <mml:mi>t</mml:mi>\n <mml:mn>3</mml:mn>\n </mml:msub>\n <mml:mo fence=\"false\" stretchy=\"false\">|<!-- | --></mml:mo>\n <mml:mo>,</mml:mo>\n <mml:mo fence=\"false\" stretchy=\"false\">|<!-- | --></mml:mo>\n <mml:msub>\n <mml:mi>t</mml:mi>\n <mml:mn>3</mml:mn>\n </mml:msub>\n <mml:mo>+</mml:mo>\n <mml:msub>\n <mml:mi>t</mml:mi>\n <mml:mn>1</mml:mn>\n </mml:msub>\n <mml:mo fence=\"false\" stretchy=\"false\">|<!-- | --></mml:mo>\n <mml:mo>≥<!-- ≥ --></mml:mo>\n <mml:mn>1</mml:mn>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">\\lvert t_1 + t_2 \\rvert , \\lvert t_2 + t_3 \\rvert , \\lvert t_3 + t_1 \\rvert \\ge 1</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> and <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"StartAbsoluteValue t 1 plus t 2 plus t 3 EndAbsoluteValue greater-than-or-equal-to 1\">\n <mml:semantics>\n <mml:mrow>\n <mml:mo fence=\"false\" stretchy=\"false\">|<!-- | --></mml:mo>\n <mml:msub>\n <mml:mi>t</mml:mi>\n <mml:mn>1</mml:mn>\n </mml:msub>\n <mml:mo>+</mml:mo>\n <mml:msub>\n <mml:mi>t</mml:mi>\n <mml:mn>2</mml:mn>\n </mml:msub>\n <mml:mo>+</mml:mo>\n <mml:msub>\n <mml:mi>t</mml:mi>\n <mml:mn>3</mml:mn>\n </mml:msub>\n <mml:mo fence=\"false\" stretchy=\"false\">|<!-- | --></mml:mo>\n <mml:mo>≥<!-- ≥ --></mml:mo>\n <mml:mn>1</mml:mn>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">\\lvert t_1 + t_2 + t_3 \\rvert \\ge 1</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> with <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"sigma 1 comma sigma 2 comma sigma 3 greater-than negative upper K\">\n <mml:semantics>\n <mml:mrow>\n <mml:msub>\n <mml:mi>σ<!-- σ --></mml:mi>\n <mml:mn>1</mml:mn>\n </mml:msub>\n <mml:mo>,</mml:mo>\n <mml:msub>\n <mml:mi>σ<!-- σ --></mml:mi>\n <mml:mn>2</mml:mn>\n </mml:msub>\n <mml:mo>,</mml:mo>\n <mml:msub>\n <mml:mi>σ<!-- σ --></mml:mi>\n <mml:mn>3</mml:mn>\n </mml:msub>\n <mml:mo>></mml:mo>\n <mml:mo>−<!-- − --></mml:mo>\n <mml:mi>K</mml:mi>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">\\sigma _1 , \\sigma _2, \\sigma _3 > -K</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> and <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"sigma 1 plus sigma 2 comma sigma 2 plus sigma 3 comma sigma 3 plus sigma 1 greater-than 1 minus upper K\">\n <mml:semantics>\n <mml:mrow>\n <mml:msub>\n <mml:mi>σ<!-- σ --></mml:mi>\n <mml:mn>1</mml:mn>\n </mml:msub>\n <mml:mo>+</mml:mo>\n <mml:msub>\n <mml:mi>σ<!-- σ --></mml:mi>\n <mml:mn>2</mml:mn>\n </mml:msub>\n <mml:mo>,</mml:mo>\n <mml:msub>\n <mml:mi>σ<!-- σ --></mml:mi>\n <mml:mn>2</mml:mn>\n </mml:msub>\n <mml:mo>+</mml:mo>\n <mml:msub>\n <mml:mi>σ<!-- σ --></mml:mi>\n <mml:mn>3</mml:mn>\n </mml:msub>\n <mml:mo>,</mml:mo>\n <mml:msub>\n <mml:mi>σ<!-- σ --></mml:mi>\n <mml:mn>3</mml:mn>\n </mml:msub>\n <mml:mo>+</mml:mo>\n <mml:msub>\n <mml:mi>σ<!-- σ --></mml:mi>\n <mml:mn>1</mml:mn>\n </mml:msub>\n <mml:mo>></mml:mo>\n <mml:mn>1</mml:mn>\n <mml:mo>−<!-- − --></mml:mo>\n <mml:mi>K</mml:mi>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">\\sigma _1 +\\sigma _2, \\sigma _2 + \\sigma _3, \\sigma _3 + \\sigma _1 > 1-K</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>, where <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper K\">\n <mml:semantics>\n <mml:mi>K</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">K</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> is a positive integer, from bounds for the Hurwitz zeta function which are shown by Bourgain’s bounds for exponential sums.</p>","PeriodicalId":106316,"journal":{"name":"Proceedings of the American Mathematical Society, Series B","volume":"70 1","pages":"0"},"PeriodicalIF":0.0000,"publicationDate":"2023-02-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Proceedings of the American Mathematical Society, Series B","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1090/bproc/142","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
In the present paper, we give bounds for the Tornheim double zeta function T(s1,s2,s3)T(s_1,s_2,s_3) when |t1|,|t2|,|t3|≥1\lvert t_1 \rvert , \lvert t_2 \rvert , \lvert t_3 \rvert \ge 1, |t1+t2|,|t2+t3|,|t3+t1|≥1\lvert t_1 + t_2 \rvert , \lvert t_2 + t_3 \rvert , \lvert t_3 + t_1 \rvert \ge 1 and |t1+t2+t3|≥1\lvert t_1 + t_2 + t_3 \rvert \ge 1 with σ1,σ2,σ3>−K\sigma _1 , \sigma _2, \sigma _3 > -K and σ1+σ2,σ2+σ3,σ3+σ1>1−K\sigma _1 +\sigma _2, \sigma _2 + \sigma _3, \sigma _3 + \sigma _1 > 1-K, where KK is a positive integer, from bounds for the Hurwitz zeta function which are shown by Bourgain’s bounds for exponential sums.