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Irrationality of values of L LL ‐functions of Dirichlet characters

机译：价值观的非理性 L L L. - dirichlet字符的功能

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Abstract > In this paper, we prove irrationality and linear independence results on values at large positive integers of <mat:math display="inline" altimg="urn:x-wiley:00246107:media:jlms12290:jlms12290-math-0003" wiley:location="equation/jlms12290-math-0003.png"> <mat:mi>L</mat:mi> </mat:math> ‐functions of Dirichlet characters. We generalize to this setting, in an optimal way, the known results on the Riemann zeta function: the Ball–Rivoal theorem and the one proved recently with Sprang and Zudilin (namely, at least <mat:math display="inline" altimg="urn:x-wiley:00246107:media:jlms12290:jlms12290-math-0004" wiley:location="equation/jlms12290-math-0004.png"> <mat:msup> <mat:mn>2</mat:mn> <mat:mrow> <mat:mrow> <mat:mo stretchy="false">(</mat:mo> <mat:mn>1</mat:mn> <mat:mo>?</mat:mo> <mat:mi>ε</mat:mi> <mat:mo stretchy="false">)</mat:mo> </mat:mrow> <mat:mfrac> <mat:mrow> <mat:mi>log</mat:mi> <mat:mi>a</mat:mi> </mat:mrow> <mat:mrow> <mat:mi>log</mat:mi> <mat:mi>log</mat:mi> <mat:mi>a</mat:mi> </mat:mrow> </mat:mfrac> </mat:mrow> </mat:msup> </mat:math> values of the Riemann zeta function at odd integers between 3 and <mat:math display="inline" altimg="urn:x-wiley:00246107:media:jlms12290:jlms12290-math-0005" wiley:location="equation/jlms12290-math-0005.png"> <mat:mi>a</mat:mi> </mat:math> are irrational if <mat:math display="inline" altimg="urn:x-wiley:00246107:media:jlms12290:jlms12290-math-0006" wiley:location="equation/jlms12290-math-0006.png"> <mat:mi>a</mat:mi> </mat:math> is large enough in terms of <mat:math display="inline" altimg="urn:x-wiley:00246107:media:jlms12290:jlms12290-math-0007" wiley:location="equation/jlms12290-math-0007.png"> <mat:mrow> <mat:mi>ε</mat:mi> <mat:mo></mat:mo> <mat:mn>0</mat:mn> </mat:mrow> </mat:math> ). >The difficult point is that positivity arguments do not apply anymore, so we have to use a completely different (and less elementary) strategy. We relate our construction to a Padé approximation problem, and use a generalization of Shidlovsky's lemma to apply Siegel's linear independence criterion. > Our results apply also to the values <mat:math display="inline" altimg="urn:x-wiley:00246107:media:jlms12290:jlms12290-math-0008" wiley:location="equation/jlms12290-math-0008.png"> <mat:mrow> <mat:mi>ζ</mat:mi> <mat:mo stretchy="false">(</mat:mo> <mat:mi>s</mat:mi> <mat:mo>,</mat:mo> <mat:mi>r</mat:mi> <mat:mo stretchy="false">)</mat:mo> </mat:mrow> </mat:math> of the Hurwitz zeta function, when <mat:math display="inline" altimg="urn:x-wiley:00246107:media:jlms12290:jlms12290-math-0009" wiley:location="equation/jlms12290-math-0009.png"> <mat:mi>r</mat:mi> </mat:math> is a fixed rational number. In the special case of the Riemann zeta function, we refine the result proved recently with Sprang and?Zudilin. </abstract> </span> <span class="z_kbtn z_kbtnclass hoverxs" style="display: none;">展开▼</span> </div> <div class="translation abstracttxt"> <span class="zhankaihshouqi fivelineshidden" id="abstract"> <span>机译：</span><Abstract Type =“Main”XML：Lang =“en”> <标题类型=“main”>抽象</ title> > 在本文中，我们证明了对大正整数的值的非理性和线性独立性结果 <mat：数学显示=“内联”Altimg =“URN：X-Wiley：00246107：媒体：JLMS12290：JLMS12290-Math-0003”Wiley：Location =“等式/ jlms12290-math-0003.png”> <MAT：MI> L </ MAT：MI> </ mat：math> - dirichlet字符的功能。我们以最佳的方式概括了这种环境，riemann Zeta函数的已知结果：球体定理和最近用Sprang和Zudilin证明的球体（即，至少 <mat：数学显示=“内联”Altimg =“URN：X-Wiley：00246107：媒体：jlms12290：jlms12290-math-0004”Wiley：location =“等式/ jlms12290-math-0004.png”> <mat：msup> <MAT：MN> 2 </ MAT：MN> <MAT：MROW> <MAT：MROW> <mat：mo stractry =“false”>（</ mat：mo> <MAT：MN> 1 </ MAT：MN> <mat：mo>？</ mat：mo> <mat：mi>ε</ mat：mi> <mat：mo stractry =“假”>）</ mat：mo> </ mat：MROW> <MAT：MFRAC> <MAT：MROW> <mat：mi> log </ mat：mi> <mat：mi> a </ mat：mi> </ mat：MROW> <MAT：MROW> <mat：mi> log </ mat：mi> <mat：mi> log </ mat：mi> <mat：mi> a </ mat：mi> </ mat：MROW> </ mat：mfrac> </ mat：MROW> </ mat：msup> </ mat：math> riemann zeta函数在3到3之间的奇数整数的值 <mat：数学显示=“内联”Altimg =“URN：X-Wiley：00246107：媒体：JLMS12290：JLMS12290-Math-0005”Wiley：Location =“等式/ jlms12290-math-0005.png”> <mat：mi> a </ mat：mi> </ mat：math> 是非理性的if. <mat：数学显示=“内联”Altimg =“URN：X-Wiley：00246107：媒体：jlms12290：jlms12290-math-0006”Wiley：location =“等式/ jlms12290-math-0006.png”> <mat：mi> a </ mat：mi> </ mat：math> 足够大 <mat：数学显示=“内联”Altimg =“URN：X-Wiley：00246107：媒体：jlms12290：jlms12290-math-0007”Wiley：location =“等式/ jlms12290-math-0007.png”> <MAT：MROW> <mat：mi>ε</ mat：mi> <mat：mo>＆ </ mat：mo> <MAT：MN> 0 </ MAT：MN> </ mat：MROW> </ mat：math> ）。 </ p> >难点是阳性参数不再适用，因此我们必须使用完全不同的（和更少的基本）策略。我们将建设与Padé近似问题相关联，并使用Shidlovsky的LEMMA的概括来应用Siegel的线性独立标准。</ p> > 我们的结果也适用于价值观 <mat：数学显示=“内联”Altimg =“URN：X-Wiley：00246107：媒体：JLMS12290：JLMS12290-Math-0008”Wiley：Location =“等式/ JLMS12290-Math-0008.png”> <MAT：MROW> <mat：mi>ζ</ mat：mi> <mat：mo stractry =“false”>（</ mat：mo> <MAT：MI> S </ MAT：MI> <mat：mo>，</ mat：mo> <MAT：MI> R </ MAT：MI> <mat：mo stractry =“假”>）</ mat：mo> </ mat：MROW> </ mat：math> urwitz zeta函数，何时 <mat：数学显示=“内联”Altimg =“URN：X-Wiley：00246107：媒体：JLMS12290：JLMS12290-Math-0009”Wiley：Location =“等式/ JLMS12290-Math-0009.png”> <MAT：MI> R </ MAT：MI> </ mat：math> 是一个固定的理性数字。在riemann Zeta功能的特殊情况下，我们最近通过sprang和zudilin来改进结果。 </ p> </摘要> </span> <span class="z_kbtn z_kbtnclass hoverxs" style="display: none;">展开▼</span> </div> </div> <div class="record"> <h2 class="all_title" id="enpatent33" >著录项</h2> <ul> <li> <span class="lefttit">来源</span> <div style="width: 86%;vertical-align: text-top;display: inline-block;"> <a href='/journal-foreign-31023/'>《The Journal of the London Mathematical Society》</a> <b style="margin: 0 2px;">|</b><span>2020年第2期</span><b style="margin: 0 2px;">|</b><span>共20页</span> </div> </li> <li> <div class="author"> <span class="lefttit">作者</span> <p id="fAuthorthree" class="threelineshidden zhankaihshouqi"> <a href="/search.html?doctypes=4_5_6_1-0_4-0_1_2_3_7_9&sertext=Fischler Stéphane&option=202" target="_blank" rel="nofollow">Fischler Stéphane;</a> </p> <span class="z_kbtnclass z_kbtnclassall hoverxs" id="zkzz" style="display: none;">展开▼</span> </div> </li> <li> <div style="display: flex;"> <span class="lefttit">作者单位</span> <div style="position: relative;margin-left: 3px;max-width: 639px;"> <div class="threelineshidden zhankaihshouqi" id="fOrgthree"> <p>Laboratoire de Mathématiques d'OrsayUniversité Paris‐Saclay91405 Orsay France;</p> </div> <span class="z_kbtnclass z_kbtnclassall hoverxs" id="zhdw" style="display: none;">展开▼</span> </div> </div> </li> <li > <span class="lefttit">收录信息</span> <span style="width: 86%;vertical-align: text-top;display: inline-block;"></span> </li> <li> <span class="lefttit">原文格式</span> <span>PDF</span> </li> <li> <span class="lefttit">正文语种</span> <span>eng</span> </li> <li> <span class="lefttit">中图分类</span> <span><a href="https://www.zhangqiaokeyan.com/clc/156.html" title="数学">数学;</a></span> </li> <li class="antistop"> <span class="lefttit">关键词</span> <p style="width: 86%;vertical-align: text-top;"> <a style="color: #3E7FEB;" href="/search.html?doctypes=4_5_6_1-0_4-0_1_2_3_7_9&sertext=11J72 (primary)&option=203" rel="nofollow">11J72 (primary);</a> <a style="color: #3E7FEB;" href="/search.html?doctypes=4_5_6_1-0_4-0_1_2_3_7_9&sertext=11M06&option=203" rel="nofollow">11M06;</a> <a style="color: #3E7FEB;" href="/search.html?doctypes=4_5_6_1-0_4-0_1_2_3_7_9&sertext=11M35&option=203" rel="nofollow">11M35;</a> <a style="color: #3E7FEB;" href="/search.html?doctypes=4_5_6_1-0_4-0_1_2_3_7_9&sertext=33C20 (secondary)&option=203" rel="nofollow">33C20 (secondary);</a> </p> <div class="translation"> 机译：11J72（初级）;11M06;11M35;33C20（二级）; </div> </li> </ul> </div> </div> <div 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initial;height: auto;word-break: break-all;white-space: initial;text-overflow: initial;overflow: initial;"> <span>机译：外生菌根垫和非垫森林土壤的微生物和生化动力学。</span> </p> </li> <li> <div> <b>8. </b><a class="enjiyixqcontent" href="/academic-journal-foreign-pmc_detail_thesis/040001861150.html">RNA editing of mat-r transcripts in maize and soybean increases similarity of the encoded protein to fungal and bryophyte group II intron maturases: evidence that mat-r encodes a functional protein.</a> <b>[O] </b> . <span> <a href="/search.html?doctypes=4_5_6_1-0_4-0_1_2_3_7_9&sertext=M C Thomson&option=202" target="_blank" rel="nofollow" class="tuijian_auth tuijian_authcolor">M C Thomson,</a> <a href="/search.html?doctypes=4_5_6_1-0_4-0_1_2_3_7_9&sertext=J L Macfarlane&option=202" target="_blank" rel="nofollow" class="tuijian_auth tuijian_authcolor">J L Macfarlane,</a> <a href="/search.html?doctypes=4_5_6_1-0_4-0_1_2_3_7_9&sertext=C T Beagley&option=202" target="_blank" rel="nofollow" class="tuijian_auth tuijian_authcolor">C T Beagley,</a> <span>1994</span> </span> </div> <p class="zwjiyix translation" style="max-width: initial;height: auto;word-break: break-all;white-space: initial;text-overflow: initial;overflow: initial;"> <span>机译：玉米和大豆中mat-r转录本的RNA编辑增加了编码的蛋白质与真菌和苔藓类植物群II内含子成熟酶的相似性：证据表明mat-r编码功能性蛋白质。</span> </p> </li> <li> <div> <b>9. </b><a class="enjiyixqcontent" href="/open-access_resources_thesis/01000126909817.html">Fabrication of <inline-formula><math altimg="none" display="inline" overflow="scroll"><mrow><msup><mi mathvariant="normal">Yb</mi><mrow><mn>3</mn><mo>+</mo></mrow></msup><mo>:</mo><msup><mi mathvariant="normal">Er</mi><mrow><mn>3</mn><mo>+</mo></mrow></msup></mrow></math></inline-formula> co-doped <inline-formula><math altimg="none" display="inline" overflow="scroll"><mrow><msub><mi mathvariant="normal">Al</mi><mn>2</mn></msub><msub><mi mathvariant="normal">O</mi><mn>3</mn></msub></mrow></math></inline-formula> ridge waveguides by the dry etching</a> <b>[O] </b> . <span> <a href="/search.html?doctypes=4_5_6_1-0_4-0_1_2_3_7_9&sertext=Qi Song&option=202" target="_blank" rel="nofollow" class="tuijian_auth tuijian_authcolor">Qi Song </a> <span>2007</span> </span> </div> <p class="zwjiyix translation" style="max-width: initial;height: auto;word-break: break-all;white-space: initial;text-overflow: initial;overflow: initial;"> <span>机译：<inline-公式> <math altimg =“无”显示=“内联”overflow =“滚动”> <mrow> <msup> <mi mathvariant =“normal”> yb </ mi> <mrow> <mn> 3 </ mn> <mo> + </ mo> </ mrow> </ msup> <mo>：</ mo> <msup> <mi mathvariant =“正常”> er </ mi> <mrow> <mn > 3 </ mn> <mo> + </ mo> </ mrow> </ msup> </ mrow> </ math> </ inline-aruem>共掺杂<内联公式> <math Altimg =“无“显示=”内联“溢出=”滚动“> <mrow> <msub> <mi mathvariant =”普通“> al </ mi> <mn> 2 </ mn> </ msub> <msub> <mi mathvariant = “正常”> O </ mi> <Mn> 3 </ Mn> </ msub> </ mrow> </ math> </ inline-甲术>脊波导通过干蚀刻</span> </p> </li> </ul> <ul style="display: none;"> <li> <div> <b>1. </b><a class="enjiyixqcontent" href="/academic-journal-cn_detail_thesis/0201296451408.html">Antimicrobial Activity of Minocycline-Loaded Genipin-Crosslinked Nano-Fibrous Chitosan Mats for Guided Tissue Regeneration</a> <b>[J]</b> <span> <a href="/search.html?doctypes=4_5_6_1-0_4-0_1_2_3_7_9&sertext=Peter A. Norowski&option=202" target="_blank" rel="nofollow" class="tuijian_auth tuijian_authcolor"> . Peter A. Norowski</a> <a href="/search.html?doctypes=4_5_6_1-0_4-0_1_2_3_7_9&sertext=Jegdish Babu&option=202" target="_blank" rel="nofollow" class="tuijian_auth tuijian_authcolor">,Jegdish Babu</a> <a href="/search.html?doctypes=4_5_6_1-0_4-0_1_2_3_7_9&sertext=Pradeep C. Adatrow&option=202" target="_blank" rel="nofollow" class="tuijian_auth tuijian_authcolor">,Pradeep C. Adatrow</a> <span> <a href="/journal-cn-56591/" target="_blank" rel="nofollow" class="tuijian_authcolor"> . 生物材料与纳米技术（英文） </a> </span> <span> . 2012</span><span>,第4期</span> </span> </div> </li> <li> <div> <b>2. </b><a class="enjiyixqcontent" href="/academic-journal-cn_china-communications_thesis/0201280142009.html">Energy Efficiency Optimization for Heterogeneous Cellular Networks Modeled by Matérn Hard-Core Point Process</a> <b>[J]</b> <span> <a href="/search.html?doctypes=4_5_6_1-0_4-0_1_2_3_7_9&sertext=Yonghong Chen&option=202" target="_blank" rel="nofollow" class="tuijian_auth tuijian_authcolor"> . Yonghong Chen</a> <a href="/search.html?doctypes=4_5_6_1-0_4-0_1_2_3_7_9&sertext=Jie Yang&option=202" target="_blank" rel="nofollow" class="tuijian_auth tuijian_authcolor">,Jie Yang</a> <a href="/search.html?doctypes=4_5_6_1-0_4-0_1_2_3_7_9&sertext=Xuehong Cao&option=202" target="_blank" rel="nofollow" class="tuijian_auth tuijian_authcolor">,Xuehong Cao</a> <span> <a href="/journal-cn-8812/" target="_blank" rel="nofollow" class="tuijian_authcolor"> . 中国通信 </a> </span> <span> . 2020</span><span>,第008期</span> </span> </div> </li> <li> <div> <b>3. </b><a class="enjiyixqcontent" href="/academic-journal-cn_detail_thesis/0201296549508.html">Preparation and Characterization of Aluminium Wire Mat Reinforced Polyester Composite</a> <b>[J]</b> <span> <a href="/search.html?doctypes=4_5_6_1-0_4-0_1_2_3_7_9&sertext=Rajib Chandra Das&option=202" target="_blank" rel="nofollow" class="tuijian_auth tuijian_authcolor"> . Rajib Chandra Das</a> <a href="/search.html?doctypes=4_5_6_1-0_4-0_1_2_3_7_9&sertext=Arif Ul Islam&option=202" target="_blank" rel="nofollow" class="tuijian_auth tuijian_authcolor">,Arif Ul Islam</a> <a href="/search.html?doctypes=4_5_6_1-0_4-0_1_2_3_7_9&sertext=Sanchita Dewanjee&option=202" target="_blank" rel="nofollow" class="tuijian_auth tuijian_authcolor">,Sanchita Dewanjee</a> <span> <a href="/journal-cn-56222/" target="_blank" rel="nofollow" class="tuijian_authcolor"> . 材料物理与化学进展（英文） </a> </span> <span> . 2019</span><span>,第4期</span> </span> </div> </li> <li> <div> <b>4. </b><a class="enjiyixqcontent" href="/academic-conference-cn_meeting-67144_thesis/020223052285.html">电导率各向异性组织中MAT-MI声源强度分布</a> <b>[C]</b> <span> <a href="/search.html?doctypes=4_5_6_1-0_4-0_1_2_3_7_9&sertext=YU Zheng-feng&option=202" target="_blank" rel="nofollow" class="tuijian_auth tuijian_authcolor"> . YU Zheng-feng</a> <a href="/search.html?doctypes=4_5_6_1-0_4-0_1_2_3_7_9&sertext=余正风&option=202" target="_blank" rel="nofollow" class="tuijian_auth tuijian_authcolor">,余正风</a> <a href="/search.html?doctypes=4_5_6_1-0_4-0_1_2_3_7_9&sertext=LI Yu-zhi&option=202" target="_blank" rel="nofollow" class="tuijian_auth tuijian_authcolor">,LI Yu-zhi</a> <span> <a href="/conference-cn-67144/" target="_blank" rel="nofollow" class="tuijian_authcolor"> . 2018年全国声学大会 </a> <span> <span> . 2018</span> </span> </div> </li> <li> <div> <b>5. </b><a class="enjiyixqcontent" href="/academic-degree-domestic_mphd_thesis/02031264616.html">豆荚结构对大豆食心虫Leguminivora glycinivorella(Mats)抗性机制的研究</a> <b>[A] </b> <span> <a href="/search.html?doctypes=4_5_6_1-0_4-0_1_2_3_7_9&sertext=吕德东&option=202" target="_blank" rel="nofollow" class="tuijian_auth tuijian_authcolor"> . 吕德东</a> <span> . 2018</span> </span> </div> </li> </ul> <ul style="display: none;"> <li> <div> <b>1. </b><a class="enjiyixqcontent" href="/patent-detail/06130454705050.html">KNOCKDOWN MAT WITH TATAMI-MAT PATTERN, MAT FOR INTERMEDIATE SECTION, MAT FOR EDGE AND MAT FOR CORNER</a> <b>[P]</b> . <span> 外国专利：  JPH10152977A </span> <span> . 1998-06-09</span> </div> <p class="zwjiyix translation" style="max-width: initial;height: auto;word-break: break-all;white-space: initial;text-overflow: initial;overflow: initial;"> <span>机译：带有TATAMI-MAT模式的Knockdown MAT，中间部分的MAT，边缘的MAT和角落的MAT </span> </p> </li> <li> <div> <b>2. </b><a class="enjiyixqcontent" href="/patent-detail/06130449686327.html">The tatami mat table which uses the tatami mat thread and the said tatami mat thread for the tatami mat table</a> <b>[P]</b> . <span> 外国专利：  JP3081371U </span> <span> . 2001-11-02</span> </div> <p class="zwjiyix translation" style="max-width: initial;height: auto;word-break: break-all;white-space: initial;text-overflow: initial;overflow: initial;"> <span>机译：将榻榻米线和所述榻榻米线用于榻榻米桌的榻榻米桌 </span> </p> </li> <li> <div> <b>3. </b><a class="enjiyixqcontent" href="/patent-detail/06130416520598.html">Producing preform, comprises providing fiber semifinished product formed as fiber mat comprising two layers, clamping fiber mat, in which individual layers of fiber mat are separately clamped, and forming fiber mat using forming device</a> <b>[P]</b> . <span> 外国专利：  DE102012021738A1 </span> <span> . 2014-04-30</span> </div> <p class="zwjiyix translation" style="max-width: initial;height: auto;word-break: break-all;white-space: initial;text-overflow: initial;overflow: initial;"> <span>机译：生产预型件，包括提供形成为包括两层的纤维垫的纤维半成品，夹持纤维垫，其中将纤维垫的各层分别夹持，以及使用成形装置成形纤维垫 </span> </p> </li> </ul> </div> </div> </div> <div class="theme cardcommon" style="overflow: auto;display:none"> <h3 class="all_title" id="enpatent55">相关主题</h3> <ul id="subject"> </ul> </div> </div> </div> </div> <div class="right rightcon"> <div class="details_img cardcommon clearfix" style="margin-bottom: 10px;display:none;" > </div> </div> </div> <div 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