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History,人物,哈代,戈弗雷·哈罗德·哈代,纯数学研究,数学合作,MHuiG, @MHuiG, Blog, 博客, Magicland, 魔法世界"><meta desc="" name="description" content="介绍了英国数学家戈弗雷·哈罗德·哈代的生平经历、学术贡献及个人特点。涵盖其教育背景、与李特尔伍德和拉马努金的著名合作、在数论和解析领域的成就，以及他的晚年生活。 - MHuiG - Magicland"><meta property="og:type" content="website"><meta property="og:title" content="Magicland"><meta property="og:url" content="https://blog.mhuig.top/notes/person/12"><meta property="og:site_name" content="Magicland"><meta property="og:description" content="介绍了英国数学家戈弗雷·哈罗德·哈代的生平经历、学术贡献及个人特点。涵盖其教育背景、与李特尔伍德和拉马努金的著名合作、在数论和解析领域的成就，以及他的晚年生活。"><meta property="og:locale"><meta property="og:image" content="https://blog.mhuig.top/lib/favicon/android-chrome-192x192.png"><meta property="article:published_time" content="2026-05-09T01:07:00.000Z"><meta property="article:modified_time" content="2026-05-09T01:10:00.000Z"><meta property="article:author" content="MHuiG"><meta property="article:tag" content="History"><meta property="article:tag" content="人物"><meta property="article:tag" 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itemscope="" itemtype="http://schema.org/CreativeWork"><meta itemprop="name" content="Hardy数学家哈代的生平、贡献与轶事"><meta itemprop="description" content="介绍了英国数学家戈弗雷·哈罗德·哈代的生平经历、学术贡献及个人特点。涵盖其教育背景、与李特尔伍德和拉马努金的著名合作、在数论和解析领域的成就，以及他的晚年生活。"></span><span hidden=""><meta itemprop="image" content="/lib/favicon/android-chrome-192x192.png"></span><div class="article-meta" id="top"><span hidden="" itemprop="name headline"></span></div><div id="layoutHelper-page-plugins"></div><div id="post-body" itemprop="articleBody"><p> <span class="p logo center large">Hardy</span></p><br><h1 hidden="">Hardy</h1><div class="story post-story"><h2 id="生平"><a href="#生平" class="headerlink" title="生平"></a>生平</h2><p>戈弗雷・哈罗德・哈代（Godfrey Harold Hardy，1877-1947），出生于英国克兰利。他的父亲艾萨克・哈代（Isaac Hardy）是克兰利学校（Cranleigh School）的财务主管和美术教师。他母亲索菲亚（Sophia）曾在林肯教师培训学校（Lincoln Teacher's Training School）任教。两位父母都非常聪明，具备一定的数学才能，但由于出身贫寒，没能接受大学教育。</p><p>Hardy 在克兰利学校就读至十二岁，并取得了优异的成绩。</p><p>Hardy 似乎没有像许多年轻数学家那样，对数学怀有浓厚的兴趣。他在自传中写道：“我不记得自己在少年时期对数学有任何激情，而我对数学家的职业的理解也相当平淡。我认为数学就是考试和奖学金的工具：我想要击败其他男孩，而数学似乎是让我最有把握做到这一点的途径。”</p><p>他在 1889 年赢得了温彻斯特公学（Winchester College）的奖学金，并于次年入学。温彻斯特公学是英国最优秀的数学训练学校。</p><p>虽然 Hardy 在那里的教育受益匪浅，但他并不喜欢学校里除学术以外的任何训练。值得注意的是，尽管他对球类运动尤其是板球有着浓厚的兴趣，但在温彻斯特公学他从未接受过任何体育训练。</p><p>在温彻斯特公学学习期间，Hardy 获得了剑桥大学三一学院（Trinity College, Cambridge）的公开奖学金，并于 1896 年入学。</p><p>在剑桥，Hardy 被分配给了最著名的导师罗伯特・拉姆齐・韦伯（Robert Rumsey Webb, 1850-1936）。他很快意识到，Webb 并不关心数学的本质，只关心应试技巧来取得考试中的最高分数（从 1865 年到 1909 年，他训练了 100 名学生登上了 Wrangler 前 10 名）。</p><p>Hardy 一度考虑转而学习历史。然而，他最终换了导师，改为洛夫（Augustus Edward Hough Love, 1863-1940）。</p><p>Hardy 在他自传中表达了对 Love 的感激之情：是 Love 教授第一次打开了我的眼界，他教会了我一些术语，并让我第一次认真理解了分析学的概念。但是我最感激的是他建议我阅读 Jordan 的《分析教程》（Cours d'analyse）；我永远不会忘记读到这本著作时的震撼，它是我们这一代许多数学家的启蒙之作，我第一次在阅读过程中明白了数学的真正意义。</p><p>约当（Marie Ennemond Camille Jordan，1838-1922），法国数学家。以他名字命名的有 Jordan 曲线定理（Jordan curve theorem）、Jordan 矩阵（Jordan matrix）、Jordan 测度（Jordan measure）等。著有《Cours d'analyses》。<br>线性代数中 Gauss-Jordan 消去法（Gauss-Jordan elimination）里的 Jordan 是 Wilhelm Jordan，Jordan 代数（Jordan algebra）里的 Jordan 是 Pascual Jordan。</p><p>1898 年，Hardy 在剑桥数学 Tripos 考试中取得了 “第四名优等生”（Fourth Wrangler）的成绩。尽管他认为这个评判系统非常愚蠢，但他仍然为未能夺冠而感到懊恼。</p><table><thead><tr><th>分类</th><th>人物</th></tr></thead><tbody><tr><td>著名的第 1 名优等生</td><td>George Airy, Arthur Cayley, Donald Coxeter, Ben Green, Littlewood, 李显龙（Lee Hsien Loong）, Frank Ramsey, Lord Rayleigh, George Stokes</td></tr><tr><td>著名的第 2 名优等生</td><td>James Clerk Maxwell, J. J. Thomson, Lord Kelvin, William Clifford, Augustus Edward Hough Love, Peter Swinnerton-Dyer</td></tr><tr><td>著名的第 3-7 名优等生</td><td>第 4 名：George Green, Hardy<br>第 7 名：Bertrand Russell</td></tr></tbody></table><p>1900 年，Hardy 被选为三一学院的研究员，随后在 1901 年，他与詹姆斯・吉恩斯（James Hopwood Jeans, 1877-1946）共同获得了 “史密斯奖”。</p><table><thead><tr><th>年份</th><th>获奖者</th><th>年份</th><th>获奖者</th></tr></thead><tbody><tr><td>1841</td><td>George Gabriel Stokes</td><td>1842</td><td>Arthur Cayley</td></tr><tr><td>1845</td><td>Lord Kelvin</td><td>1854</td><td>James Clerk Maxwell</td></tr><tr><td>1880</td><td>J. J. Thomson</td><td>1901</td><td>Hardy, Jeans</td></tr><tr><td>1912</td><td>Louis Mordell</td><td>1914</td><td>Bhupati Mohan Sen</td></tr><tr><td>1930</td><td>John Arthur Todd</td><td>1931</td><td>H. S. M. Coxeter</td></tr><tr><td>1936</td><td>Alan Turing</td><td></td><td></td></tr></tbody></table><p>部分史密斯奖获得者</p><p>Hardy 职业生涯的下一个阶段延续到 1911 年。正如伯克尔（Burkill）在人物传记中写道：“他撰写了许多关于级数和积分收敛性及相关主题的论文。虽然这些工作确立了他作为分析学家的声誉，但他在这一早期阶段对数学最重要的贡献是《纯数学教程》（A Course of Pure Mathematics, 1908）。这本书是第一本严谨的英文版关于数、函数、极限等内容的讲解，专为本科生编写，由此彻底改变了大学数学教学。”</p><p>Hardy 描述了这一时期的自己：“我写了很多东西…… 但几乎没有什么是重要的；能让我至今还能感到满意的论文不超过四五篇。”</p><p>值得注意的是，Hardy 是一个非常诚实的人，特别是在评价自己能力、优缺点时更是如此。</p><p>Hardy 的工作在 1911 年发生了重大转变，他开始了与<br>约翰・伊登斯尔・李特尔伍德（John Edensor Littlewood,<br>1885-1977）的合作，这段合作关系持续了 35 年。</p><p>接着，在 1913 年初，他收到了来自印度的拉马努金（Srinivasa Ramanujan, 1887-1920）的第一封信，开启了他第二段重要的合作关系。到 1914 年第一次世界大战爆发时，Ramanujan 已经来到了剑桥，这在艰难的战争时期对 Hardy 来说是一种安慰。</p><p>Littlewood 离开剑桥，加入皇家炮兵团服役。Hardy 也志愿参战，但因健康原因被拒绝。然而，Hardy 对战争的看法使他与剑桥的大多数同事格格不入。</p><p>他对德国怀有深厚的尊敬：他对德国有着强烈的好感。毕竟，德国一直是 19 世纪重要的教育力量。对于东欧、俄罗斯和美国，德国的大学教授了他们何为科研。在大多数方面，德国文化，包括它的社会福利体系，都比他自己的文化更为高尚。Hardy 和罗素（Bertrand Arthur William Russell,1872-1970）一样，都认为这场战争不应该爆发。此外，出于对英国政治家的根深蒂固的不信任，他认为这场战争的错误更多在英国一方。</p><p>在剑桥深感不快的 Hardy，于 1919 年接受了牛津大学萨维尔几何学教授（Savilian Professor of Geometry）的职位。这段时期是他最快乐的时光，也是他与 Littlewood 作出最优秀数学成果的阶段。尽管这段合作期间，Littlewood 在剑桥，而 Hardy 在牛津，导致合作在实际操作上相当困难。正如 Hardy 在自传中所写：我在四十出头时状态最佳，那时我在牛津担任教授。</p><p>在剑桥期间，他选择不住在最好的房间里，Hilbert 甚至担心 Hardy 没有得到应有的待遇，写信给学院院长，指出 “英国最好的数学家” 应该住在最好的房间里。然而，Hardy 并不在意这些。他在 1924 至 1926 年期间担任科学工作者协会主席。那时，在政治立场分化严重的情况下，Hardy 对美国和苏联抱有同样的好感。他在 1928-29 学年赴普林斯顿交流。</p><p>尽管他在剑桥期间并不快乐，但 1931 年在霍布森（Hobson）退休后，Hardy 回到剑桥担任萨德莱尔（Sadleirian）数学教授。Hardy 回到剑桥有两个原因：一是他始终认为剑桥是英国数学的中心，Sadleirian 数学教授是英国数学界最重要的职位；二是他可以在剑桥学院保留自己的房间，而在牛津却无法做到。对于未婚的 Hardy 来说，这在他步入老年时具有特别的吸引力。</p><p>Hardy 的研究兴趣涵盖纯数学的多个领域 —— 丢番图分析（Diophantine analysis）、发散级数的求和、傅里叶级数（Fourier series）、黎曼<mjx-container class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align:-.464ex" xmlns="http://www.w3.org/2000/svg" width="1.072ex" height="2.041ex" role="img" focusable="false" viewBox="0 -697 474 902"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="\zeta"><g data-mml-node="mi" data-latex="\zeta"><path data-c="1D701" d="M276-175C251-175 228-165 206-146 200-141 195-139 191-139 182-139 178-144 178-153 178-158 182-165 191-173 215-194 243-205 276-205 337-205 387-142 387-81 387-32 355-1 315 16 302 20 185 62 182 63 133 84 108 128 108 195 108 262 129 334 172 410 215 486 265 540 322 571 335 560 355 554 383 554 444 554 474 566 474 589 474 610 447 620 392 620 368 620 348 617 331 611 328 621 326 632 326 643 326 652 328 665 331 680 331 691 326 697 315 697 300 697 293 679 293 644 293 626 296 610 303 597 233 560 173 499 122 414 71 329 46 247 46 168 46 92 76 36 122 9 161-10 190-22 209-29 248-46 294-52 315-80 323-89 327-101 327-114 327-144 306-175 276-175M391 590C404 590 417 589 428 587 416 585 401 584 384 584 373 584 364 585 357 587 366 589 377 590 391 590Z"></path></g></g></g></svg></mjx-container> 函数（Riemann zeta function）以及素数分布。他与 Littlewood 长久的合作产生了最高质量的数学成果。在这段合作中，Hardy 承认 Littlewood 在数学技术上的能力更强，但同时，Hardy 也展现了自己卓越的数学洞察力，并且能够将他们的研究成果以极其清晰的方式写成论文。</p><p>更为非凡的是 Hardy 与 Ramanujan 的合作。1913 年，Ramanujan 从印度寄给 Hardy 一份手稿，Hardy 立即识别出了其中的天才之处。在此之前，另外两位顶级数学家未能识别出 Ramanujan 的才华。Hardy 将 Ramanujan 带到了剑桥，并共同撰写了五篇极为重要的论文。</p><p>Hardy 是个天生的合作者，他不仅与 Littlewood 和 Ramanujan 合作过，还与蒂奇马什（Edward Charles Titchmarsh, 1899-1963）、英厄姆（Albert Edward Ingham, 1900-1967）、朗道（Edmund Georg Hermann Landau, 1877-1938）、波利亚（George Pólya，1887-1985）、赖特（Edward Maitland Wrigh，1906-2005）、罗格辛斯基（Werner Wolfgang Rogosinski，1894-1964）和里斯（Marcel Riesz，1886-1969）共同发表了许多论文。</p><p>除了数学之外，Hardy 唯一的热情便是板球。实际上，他的大部分生活都围绕板球展开，至少在板球赛季期间，他的一天通常是：早餐时阅读《泰晤士报》并研究板球比分，然后从上午 9 点到下午 1 点专注于数学研究。午饭后，他会步行到大学的板球场看比赛。傍晚时分，他缓步返回学院，享用晚餐，并在饭后喝一杯葡萄酒。当没有板球比赛时，他会通过《泰晤士报》阅读澳大利亚的板球比赛比分，下午则去打皇家网球（real tennis）。</p><p>Hardy 因其怪癖而闻名。他无法忍受拍照，已知的照片只有五张。他也讨厌镜子，进入旅馆房间的第一件事就是用毛巾把镜子盖住。他常玩一个有趣的游戏来 “愚弄上帝”（虽然他一生声称自己不信神）。</p><p>比如，他曾在前往丹麦的旅途中寄回一张明信片，声称自己已证明了黎曼猜想。他推测上帝不会允许船在回程时沉没而让他获得类似费马因 “最后定理” 而得到的名声。</p><p>1945 年二战结束时，Hardy 的健康迅速恶化。他渴望重新拥有创造力，因为那是他生活中唯一重要的东西，但他知道自己已无法恢复曾经的状态，这使他陷入深深的抑郁。</p><p>1946 年，他的身体状况变得极为虚弱，只能靠乘坐出租车出行，走几步就会气喘吁吁。1947 年初夏，Hardy 试图通过服用过量巴比妥类药物自杀，但未成功。之后，他决定静静等待死亡。他在某天早晨突然离世。</p><p>Hardy 一生获得了许多荣誉。1910 年，他被选为英国皇家学会（Royal Society）院士，并在 1920 年获得皇家奖章（Royal Medal），1940 年获得西尔维斯特奖章（Sylvester Medal），1947 年获得柯普利奖章（Copley Medal），这些奖项都是为了表彰他在纯数学多个领域的重要贡献。</p><p>他还曾两度担任伦敦数学学会主席，并在 1929 年获得该学会的德摩根奖章（De Morgan Medal）。</p></div><div class="story post-story"><h2 id="贡献"><a href="#贡献" class="headerlink" title="贡献"></a>贡献</h2><h3 id="哈代-拉马努金渐近公式（Hardy-Ramanujan-asymptotic-formula）"><a href="#哈代-拉马努金渐近公式（Hardy-Ramanujan-asymptotic-formula）" class="headerlink" title="哈代-拉马努金渐近公式（Hardy-Ramanujan asymptotic formula）"></a>哈代 - 拉马努金渐近公式（Hardy-Ramanujan asymptotic formula）</h3><p>哈代 - 拉马努金推导出了关于整数分拆数（即将一个数分解为若干整数之和的不同方法数）的渐近公式。这在数论中具有深远的意义。</p><p><mjx-container class="MathJax" jax="SVG" overflow="overflow" display="true" width="full" style="min-width:49.125ex"><svg style="vertical-align:-2.817ex;min-width:49.125ex" xmlns="http://www.w3.org/2000/svg" width="100%" height="6.765ex" role="img" focusable="false"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(0.0181,-0.0181) translate(0, -1745)"><g data-mml-node="math" data-latex="
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data-mml-node="mtd" id="mjx-eqn:1"><g data-mml-node="mtext" data-latex="\text{(1)}"><path data-c="28" d="M318-248C327-248 332-243 332-234 332-231 330-227 327-223 275-183 233-117 202-26 175 53 161 131 161 208L161 292C161 369 175 447 202 526 233 617 275 683 327 723 330 726 332 730 332 734 332 743 327 748 318 748 317 748 314 747 311 745 251 699 201 631 160 540 121 453 101 371 101 292L101 208C101 129 121 47 160-40 201-131 251-199 311-245 314-247 317-248 318-248Z"></path><path data-c="31" d="M269 666C228 624 168 603 89 603L89 564C141 564 184 572 217 588L217 82C217 64 213 52 204 47 195 42 170 39 130 39L95 39 95 0C120 2 174 3 257 3 340 3 394 2 419 0L419 39 384 39C343 39 318 42 310 47 302 52 297 64 297 82L297 636C297 660 295 666 269 666Z" transform="translate(389,0)"></path><path data-c="29" d="M78-245C138-199 188-131 229-40 268 47 288 129 288 208L288 292C288 371 268 453 229 540 188 631 138 699 78 745 75 747 72 748 71 748 62 748 57 743 57 734 57 730 59 726 62 723 114 683 156 617 187 526 214 447 228 369 228 292L228 208C228 131 214 53 187-26 156-117 114-183 62-223 59-227 57-231 57-234 57-243 62-248 71-248 72-248 75-247 78-245Z" transform="translate(889,0)"></path></g></g></g></svg></g></g></g></g></svg></mjx-container></p><h3 id="素数分布"><a href="#素数分布" class="headerlink" title="素数分布"></a>素数分布</h3><p>Hardy 和 Littlewood 在素数定理方面也做出了贡献，尤其是在证明与素数的渐近分布相关的命题上。并就素数提出了两个著名的 Hardy-Littlewood 猜想：</p><p>Hardy-Littlewood 第一猜想，也称为素数孪生猜想的推广形式。该猜想涉及了与素数配对的渐近公式。</p><p>Hardy-Littlewood 第二猜想是指：</p><p><mjx-container class="MathJax" jax="SVG" overflow="overflow" display="true" width="full" style="min-width:31.906ex"><svg style="vertical-align:-.566ex;min-width:31.906ex" xmlns="http://www.w3.org/2000/svg" width="100%" height="2.262ex" role="img" focusable="false"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(0.0181,-0.0181) translate(0, -750)"><g data-mml-node="math" data-latex="
\pi(x + y) \leq \pi(x) + \pi(y)
"><g data-mml-node="mtable" data-latex="
\pi(x + y) \leq \pi(x) + \pi(y)
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35 421 43 417 53 398 100 389 153 389 214 389 261 395 312 406 366L515 366C550 366 567 378 567 403 567 426 549 431 524 431Z"></path></g><g data-mml-node="mo" data-latex="(" transform="translate(8678.4,0)"><path data-c="28" d="M318-248C327-248 332-243 332-234 332-231 330-227 327-223 275-183 233-117 202-26 175 53 161 131 161 208L161 292C161 369 175 447 202 526 233 617 275 683 327 723 330 726 332 730 332 734 332 743 327 748 318 748 317 748 314 747 311 745 251 699 201 631 160 540 121 453 101 371 101 292L101 208C101 129 121 47 160-40 201-131 251-199 311-245 314-247 317-248 318-248Z"></path></g><g data-mml-node="mi" data-latex="y" transform="translate(9067.4,0)"><path data-c="1D466" d="M163 442C118 442 83 417 58 367 39 328 29 301 29 286 29 277 34 272 45 272 59 272 60 278 64 293 87 372 119 412 160 412 173 412 180 403 180 385 180 369 175 346 164 317 126 214 107 145 107 108 107 32 154-13 231-13 264-13 295-1 323 23 288-109 233-175 158-175 124-175 101-164 89-142 129-140 149-121 149-85 149-60 136-47 109-47 72-47 50-78 50-115 50-170 100-205 158-205 273-205 367-99 392-1L486 377C489 388 490 396 490 401 490 421 479 431 458 431 442 431 429 423 420 408 414 387 409 369 406 354L342 97C332 61 279 16 234 16 196 16 177 41 177 92 177 134 194 198 227 284 240 319 247 343 247 357 247 406 212 442 163 442Z"></path></g><g data-mml-node="mo" data-latex=")" transform="translate(9557.4,0)"><path data-c="29" d="M78-245C138-199 188-131 229-40 268 47 288 129 288 208L288 292C288 371 268 453 229 540 188 631 138 699 78 745 75 747 72 748 71 748 62 748 57 743 57 734 57 730 59 726 62 723 114 683 156 617 187 526 214 447 228 369 228 292L228 208C228 131 214 53 187-26 156-117 114-183 62-223 59-227 57-231 57-234 57-243 62-248 71-248 72-248 75-247 78-245Z"></path></g></g></g></g></svg><svg data-labels="true" preserveAspectRatio="xMaxYMid" viewBox="1278 -750 1 1000"><g data-labels="true" transform="matrix(1 0 0 -1 0 0)"><g data-mml-node="mtd" id="mjx-eqn:2"><g data-mml-node="mtext" data-latex="\text{(2)}"><path data-c="28" d="M318-248C327-248 332-243 332-234 332-231 330-227 327-223 275-183 233-117 202-26 175 53 161 131 161 208L161 292C161 369 175 447 202 526 233 617 275 683 327 723 330 726 332 730 332 734 332 743 327 748 318 748 317 748 314 747 311 745 251 699 201 631 160 540 121 453 101 371 101 292L101 208C101 129 121 47 160-40 201-131 251-199 311-245 314-247 317-248 318-248Z"></path><path data-c="32" d="M237 666C186 666 143 648 106 612 69 576 50 534 50 483 50 449 75 424 106 424 136 424 161 450 161 480 161 513 137 536 105 536 102 536 100 536 98 535 117 584 161 627 224 627 306 627 352 556 352 470 352 403 318 331 250 255L62 43C49 28 50 29 50 0L421 0 450 180 417 180C409 129 402 100 396 91 391 86 361 84 306 84L139 84 236 179C304 243 390 312 419 365 439 400 449 435 449 470 449 588 357 666 237 666Z" transform="translate(389,0)"></path><path data-c="29" d="M78-245C138-199 188-131 229-40 268 47 288 129 288 208L288 292C288 371 268 453 229 540 188 631 138 699 78 745 75 747 72 748 71 748 62 748 57 743 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117 331 68 342-11 379-11 400-11 423 9 423 30 423 35 421 43 417 53 398 100 389 153 389 214 389 261 395 312 406 366L515 366C550 366 567 378 567 403 567 426 549 431 524 431Z"></path></g><g data-mml-node="mo" data-latex="(" transform="translate(570,0)"><path data-c="28" d="M318-248C327-248 332-243 332-234 332-231 330-227 327-223 275-183 233-117 202-26 175 53 161 131 161 208L161 292C161 369 175 447 202 526 233 617 275 683 327 723 330 726 332 730 332 734 332 743 327 748 318 748 317 748 314 747 311 745 251 699 201 631 160 540 121 453 101 371 101 292L101 208C101 129 121 47 160-40 201-131 251-199 311-245 314-247 317-248 318-248Z"></path></g><g data-mml-node="mi" data-latex="x" transform="translate(959,0)"><path data-c="1D465" d="M527 373C527 419 482 442 432 442 389 442 355 419 329 373 308 419 273 442 222 442 173 442 133 419 101 374 74 335 60 306 60 287 60 278 65 273 75 273 84 273 90 278 92 287 111 345 153 413 220 413 253 413 269 392 269 351 269 330 251 252 216 118 199 51 169 18 126 18 112 18 99 21 88 26 114 36 127 54 127 80 127 106 114 119 87 119 54 119 29 91 29 58 29 12 76-11 125-11 167-11 201 12 228 58 247 12 283-11 335-11 383-11 423 12 455 57 482 96 496 125 496 144 496 153 491 158 481 158 472 158 467 153 464 144 447 87 402 18 337 18 304 18 287 38 287 79 287 92 292 120 303 165L337 300C356 375 387 413 431 413 445 413 458 410 469 405 442 396 429 378 429 351 429 325 443 312 470 312 502 312 527 341 527 373Z"></path></g><g data-mml-node="mo" data-latex=")" transform="translate(1531,0)"><path data-c="29" d="M78-245C138-199 188-131 229-40 268 47 288 129 288 208L288 292C288 371 268 453 229 540 188 631 138 699 78 745 75 747 72 748 71 748 62 748 57 743 57 734 57 730 59 726 62 723 114 683 156 617 187 526 214 447 228 369 228 292L228 208C228 131 214 53 187-26 156-117 114-183 62-223 59-227 57-231 57-234 57-243 62-248 71-248 72-248 75-247 78-245Z"></path></g></g></g></svg></mjx-container> 是素数计数函数。</p><h3 id="积分形式的哈代不等式"><a href="#积分形式的哈代不等式" class="headerlink" title="积分形式的哈代不等式"></a>积分形式的哈代不等式</h3><p>如果<mjx-container class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align:-.464ex" xmlns="http://www.w3.org/2000/svg" width="1.249ex" height="2.059ex" role="img" focusable="false" viewBox="0 -705 552 910"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="f"><g data-mml-node="mi" data-latex="f"><path data-c="1D453" d="M552 633C552 677 509 705 462 705 400 705 357 665 334 586 329 568 318 517 302 433L237 433C215 433 204 432 204 411 204 400 214 395 235 395L295 395 222 8C211-49 201-91 192-119 180-156 163-175 141-175 126-175 114-171 103-164 135-159 151-140 151-108 151-82 138-69 111-69 77-69 53-99 53-133 53-177 94-205 141-205 166-205 189-195 208-174 240-141 265-94 283-31 294 8 304 46 311 84L369 395 451 395C474 395 484 396 484 419 484 428 474 433 454 433L377 433C383 474 411 625 420 644 430 665 444 675 462 675 477 675 490 671 501 664 470 657 454 639 454 608 454 582 467 569 494 569 528 569 552 599 552 633Z"></path></g></g></g></svg></mjx-container> 是非负可测函数，以及<mjx-container class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align:-.566ex" xmlns="http://www.w3.org/2000/svg" width="1.138ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 503 1000"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="p > 1"><g data-mml-node="mi" data-latex="p"><path data-c="1D45D" d="M355 442C311 442 269 419 228 373 217 419 186 442 137 442 96 442 66 409 45 344 35 311 30 292 30 286 30 277 35 272 46 272 51 272 54 273 57 275 62 284 65 291 66 298 84 374 107 412 134 412 152 412 161 398 161 371 161 356 159 339 154 321L44-118C35-151 34-155-5-155-23-155-32-163-32-178-32-189-26-194-15-194 0-194 52-191 67-191 86-191 146-194 165-194 180-194 187-186 187-170 187-160 178-155 159-155 141-155 114-156 114-144 114-131 155 26 159 43 179 6 209-13 249-13 314-13 371 20 421 87 467 148 490 213 490 280 490 367 438 442 355 442M352 412C391 412 411 382 411 323 411 296 405 260 394 215 371 128 342 71 307 42 286 25 267 16 248 16 219 16 199 29 188 56 179 77 174 92 174 100L225 309C230 331 248 354 276 377 304 400 329 412 352 412Z"></path></g></g></g></svg><mjx-break size="4"></mjx-break><svg style="vertical-align:-.566ex" xmlns="http://www.w3.org/2000/svg" width="3.52ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 1555.8 1000"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="p > 1"><g data-mml-node="mo" data-latex=">"><path data-c="3E" d="M686 227C696 232 701 239 701 250 701 261 696 268 686 273L112 545C109 546 105 547 101 547 85 547 77 539 77 522 77 513 82 506 91 502L625 250 91-2C82-6 77-13 77-22 77-39 85-47 101-47 105-47 109-46 112-45Z"></path></g><g data-mml-node="mn" data-latex="1" transform="translate(1055.8,0)"><path data-c="31" d="M269 666C228 624 168 603 89 603L89 564C141 564 184 572 217 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\int_{0}^{\infty} \left( \frac{1}{x} \int_{0}^{x} f(t) \, dt \right)^p dx \leq \left( \frac{p}{p-1} \right)^p \int_{0}^{\infty} f^p(t) \, dt
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\int_{0}^{\infty} \left( \frac{1}{x} \int_{0}^{x} f(t) \, dt \right)^p dx \leq \left( \frac{p}{p-1} \right)^p \int_{0}^{\infty} f^p(t) \, dt
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156 617 187 526 214 447 228 369 228 292L228 208C228 131 214 53 187-26 156-117 114-183 62-223 59-227 57-231 57-234 57-243 62-248 71-248 72-248 75-247 78-245Z" transform="translate(889,0)"></path></g></g></g></g></svg></g></g></g></g></svg></mjx-container></p><h3 id="哈代–李特尔伍德极大算子（Hardy–Littlewood-maximal-operator）"><a href="#哈代–李特尔伍德极大算子（Hardy–Littlewood-maximal-operator）" class="headerlink" title="哈代–李特尔伍德极大算子（Hardy–Littlewood maximal operator）"></a>哈代–李特尔伍德极大算子（Hardy–Littlewood maximal operator）</h3><p>是一个在实分析和调和分析中具有重要意义的非线性算子。给定局部可积函数<mjx-container class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align:-.566ex" xmlns="http://www.w3.org/2000/svg" width="1.249ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 552 1000"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="f: \mathbb{R}^n \to \mathbb{C}"><g data-mml-node="mi" data-latex="f"><path data-c="1D453" d="M552 633C552 677 509 705 462 705 400 705 357 665 334 586 329 568 318 517 302 433L237 433C215 433 204 432 204 411 204 400 214 395 235 395L295 395 222 8C211-49 201-91 192-119 180-156 163-175 141-175 126-175 114-171 103-164 135-159 151-140 151-108 151-82 138-69 111-69 77-69 53-99 53-133 53-177 94-205 141-205 166-205 189-195 208-174 240-141 265-94 283-31 294 8 304 46 311 84L369 395 451 395C474 395 484 396 484 419 484 428 474 433 454 433L377 433C383 474 411 625 420 644 430 665 444 675 462 675 477 675 490 671 501 664 470 657 454 639 454 608 454 582 467 569 494 569 528 569 552 599 552 633Z"></path></g></g></g></svg><mjx-break size="4"></mjx-break><svg style="vertical-align:-.566ex" xmlns="http://www.w3.org/2000/svg" width="4.039ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 1785 1000"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="f: \mathbb{R}^n \to \mathbb{C}"><g data-mml-node="mo" data-latex=":"><path data-c="3A" 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data-mjx-texclass="ORD" transform="translate(1277.8,0)"><g data-mml-node="mi" data-latex="C"><path data-c="2102" d="M653 87C674 108 685 123 685 131 685 147 677 155 660 155 653 155 646 150 637 140 576 71 499 36 406 36 334 36 284 68 256 131 231 188 219 258 219 341 219 416 230 483 252 540 282 621 329 661 392 661 437 661 481 643 526 607 571 571 600 531 614 487 619 473 626 466 637 466 650 466 657 478 657 502L657 667C657 692 650 704 637 704 626 704 618 696 614 681 610 666 601 658 586 658 566 658 536 666 494 681 452 696 417 704 389 704 298 704 219 676 151 620 75 558 37 466 37 343 37 220 75 127 150 66 219 9 300-19 391-19 492-19 580 16 653 87M615 620 615 577C607 586 595 599 579 616L587 616C598 616 608 617 615 620M256 635C247 625 239 613 231 600 200 549 176 429 176 341 176 203 206 105 266 46 141 94 79 193 79 342 79 490 138 588 256 635Z"></path></g></g></g></g></svg></mjx-container> ，极大函数定义为：</p><p><mjx-container class="MathJax" jax="SVG" overflow="overflow" display="true" width="full" 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Mf(x) = \sup_{r>0} \frac{1}{|B_r^n(x)|} \int_{B_r^n(x)} |f(y)| dy
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Mf(x) = \sup_{r>0} \frac{1}{|B_r^n(x)|} \int_{B_r^n(x)} |f(y)| dy
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19 118 49 118 108 118 132 124 168 136 216 157 301 187 359 224 390 244 407 263 415 282 415 309 415 329 402 341 376Z"></path></g><g data-mml-node="mi" data-latex="y" transform="translate(15395.8,0)"><path data-c="1D466" d="M163 442C118 442 83 417 58 367 39 328 29 301 29 286 29 277 34 272 45 272 59 272 60 278 64 293 87 372 119 412 160 412 173 412 180 403 180 385 180 369 175 346 164 317 126 214 107 145 107 108 107 32 154-13 231-13 264-13 295-1 323 23 288-109 233-175 158-175 124-175 101-164 89-142 129-140 149-121 149-85 149-60 136-47 109-47 72-47 50-78 50-115 50-170 100-205 158-205 273-205 367-99 392-1L486 377C489 388 490 396 490 401 490 421 479 431 458 431 442 431 429 423 420 408 414 387 409 369 406 354L342 97C332 61 279 16 234 16 196 16 177 41 177 92 177 134 194 198 227 284 240 319 247 343 247 357 247 406 212 442 163 442Z"></path></g></g></g></g></svg><svg data-labels="true" preserveAspectRatio="xMaxYMid" viewBox="1278 -1468.8 1 2437.5"><g data-labels="true" transform="matrix(1 0 0 -1 0 0)"><g data-mml-node="mtd" id="mjx-eqn:4" transform="translate(0,855.8)"><text data-id-align="true"></text><g data-idbox="true" transform="translate(0,-748)"><g data-mml-node="mtext" data-latex="\text{(4)}"><path data-c="28" d="M318-248C327-248 332-243 332-234 332-231 330-227 327-223 275-183 233-117 202-26 175 53 161 131 161 208L161 292C161 369 175 447 202 526 233 617 275 683 327 723 330 726 332 730 332 734 332 743 327 748 318 748 317 748 314 747 311 745 251 699 201 631 160 540 121 453 101 371 101 292L101 208C101 129 121 47 160-40 201-131 251-199 311-245 314-247 317-248 318-248Z"></path><path data-c="34" d="M353 677C344 677 336 672 330 663L28 199 28 163 289 163 289 81C289 63 285 51 278 46 271 41 252 39 219 39L194 39 194 0C223 2 269 3 331 3 393 3 439 2 468 0L468 39 443 39C410 39 391 41 384 46 377 51 373 63 373 81L373 163 471 163 471 202 373 202 373 660C373 670 366 677 353 677M295 553 295 202 67 202Z" transform="translate(389,0)"></path><path data-c="29" d="M78-245C138-199 188-131 229-40 268 47 288 129 288 208L288 292C288 371 268 453 229 540 188 631 138 699 78 745 75 747 72 748 71 748 62 748 57 743 57 734 57 730 59 726 62 723 114 683 156 617 187 526 214 447 228 369 228 292L228 208C228 131 214 53 187-26 156-117 114-183 62-223 59-227 57-231 57-234 57-243 62-248 71-248 72-248 75-247 78-245Z" transform="translate(889,0)"></path></g></g></g></g></svg></g></g></g></g></svg></mjx-container></p><p>则有如下不等式（Hardy–Littlewood maximal inequality）：</p><p><mjx-container class="MathJax" jax="SVG" overflow="overflow" display="true" width="full" style="min-width:38.243ex"><svg style="vertical-align:-.68ex;min-width:38.243ex" xmlns="http://www.w3.org/2000/svg" width="100%" height="2.49ex" role="img" focusable="false"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(0.0181,-0.0181) translate(0, -800.3)"><g data-mml-node="math" data-latex="
\|Mf\|_{L^p(\mathbb{R}^n)} \leq C(p,n)\|f\|_{L^p(\mathbb{R}^n)}
"><g data-mml-node="mtable" data-latex="
\|Mf\|_{L^p(\mathbb{R}^n)} \leq C(p,n)\|f\|_{L^p(\mathbb{R}^n)}
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662L484 108 408 657C404 682 402 683 373 683L237 683C215 683 204 682 204 660 204 649 215 644 236 644 256 644 297 647 297 631 297 629 296 623 293 613L167 110C158 72 135 49 100 42 75 39 42 43 42 16 42 5 48 0 60 0 78 0 141 3 159 3 178 3 242 0 261 0 276 0 283 8 283 24 283 33 276 38 261 39 219 39 198 52 198 78 198 82 199 89 202 100L332 621 415 26C418 9 424 0 434 0 442 0 450 7 459 20L848 627 712 82C706 60 696 47 681 42 675 40 656 39 625 39 604 39 594 37 594 16Z"></path></g><g data-mml-node="mi" data-latex="f" transform="translate(1442,0)"><path data-c="1D453" d="M552 633C552 677 509 705 462 705 400 705 357 665 334 586 329 568 318 517 302 433L237 433C215 433 204 432 204 411 204 400 214 395 235 395L295 395 222 8C211-49 201-91 192-119 180-156 163-175 141-175 126-175 114-171 103-164 135-159 151-140 151-108 151-82 138-69 111-69 77-69 53-99 53-133 53-177 94-205 141-205 166-205 189-195 208-174 240-141 265-94 283-31 294 8 304 46 311 84L369 395 451 395C474 395 484 396 484 419 484 428 474 433 454 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307 42 286 25 267 16 248 16 219 16 199 29 188 56 179 77 174 92 174 100L225 309C230 331 248 354 276 377 304 400 329 412 352 412Z"></path></g></g><g data-mml-node="mo" data-latex="(" transform="translate(1119.7,0)"><path data-c="28" d="M318-248C327-248 332-243 332-234 332-231 330-227 327-223 275-183 233-117 202-26 175 53 161 131 161 208L161 292C161 369 175 447 202 526 233 617 275 683 327 723 330 726 332 730 332 734 332 743 327 748 318 748 317 748 314 747 311 745 251 699 201 631 160 540 121 453 101 371 101 292L101 208C101 129 121 47 160-40 201-131 251-199 311-245 314-247 317-248 318-248Z"></path></g><g data-mml-node="msup" data-latex="\mathbb{R}^n" transform="translate(1508.7,0)"><g data-mml-node="TeXAtom" data-latex="\mathbb{R}" data-mjx-texclass="ORD"><g data-mml-node="mi" data-latex="R"><path data-c="211D" d="M104 590 104 95C104 45 99 43 47 43 26 43 16 36 16 22 16 4 31 0 54 0L324 0C349 0 361 7 361 22 361 36 350 43 329 43 277 43 273 45 273 95L273 310 302 310 450 83C475 43 490 20 494 14 500 5 512 0 530 0L666 0C691 0 703 7 703 22 703 33 697 40 685 42 660 47 625 82 579 146 534 208 494 267 459 322 573 344 630 402 630 496 630 563 600 613 540 644 488 671 429 685 364 685L54 685C29 685 16 678 16 663 16 649 26 642 47 642 99 642 104 640 104 590M535 597C570 576 587 542 587 496 587 427 547 385 468 368 485 396 494 438 494 495 494 552 483 596 462 629 487 622 512 612 535 597M452 495C452 433 441 394 419 378 397 362 348 353 273 353L273 593C273 630 287 642 330 642 425 642 452 595 452 495M413 316C496 184 561 93 610 43L525 43 352 311C357 312 361 312 364 312 378 312 394 313 413 316M140 642 240 642C234 631 231 616 231 596L231 93C231 72 233 55 237 43L140 43C144 55 146 72 146 93L146 592C146 613 144 630 140 642Z"></path></g></g><g data-mml-node="mi" transform="translate(755,289) scale(0.707)" data-latex="n"><path data-c="1D45B" d="M537 137C514 58 481 18 440 18 427 18 420 28 420 47 420 61 426 84 438 115 478 224 498 296 498 333 498 403 451 442 381 442 322 442 271 416 230 363 222 407 187 442 136 442 80 442 58 390 44 345 34 313 29 294 29 287 29 278 34 273 45 273 50 273 53 274 56 276 61 285 64 292 65 299 83 375 106 413 133 413 151 413 160 399 160 371 160 358 155 331 144 290L87 63C84 50 78 24 78 19 78-1 89-11 111-11 130-11 144-1 151 19 153 24 159 49 170 92L191 181 221 295C232 318 249 341 270 365 299 397 335 413 378 413 411 413 427 391 427 348 427 310 406 234 363 120 356 102 353 87 353 74 353 25 390-11 438-11 482-11 517 14 542 64 561 104 571 131 571 144 571 153 566 158 555 158 552 158 537 149 537 137Z"></path></g></g><g data-mml-node="mo" data-latex=")" transform="translate(2737.9,0)"><path data-c="29" d="M78-245C138-199 188-131 229-40 268 47 288 129 288 208L288 292C288 371 268 453 229 540 188 631 138 699 78 745 75 747 72 748 71 748 62 748 57 743 57 734 57 730 59 726 62 723 114 683 156 617 187 526 214 447 228 369 228 292L228 208C228 131 214 53 187-26 156-117 114-183 62-223 59-227 57-231 57-234 57-243 62-248 71-248 72-248 75-247 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155 137 155 167 155 198 130 219 100 219 65 219 50 200 50 163 50 63 131-22 231-22 292-22 344 0 386 44 428 88 449 141 449 202 449 260 432 310 398 353 361 400 315 423 259 423 212 423 171 408 138 378L138 556C165 548 191 544 218 544 264 544 304 555 338 578 369 597 390 616 402 634 408 642 411 648 411 651 411 661 406 666 396 666 341 645 294 634 256 634 211 634 168 643 127 662 122 664 118 665 114 665 105 665 100 656 100 637L100 345C100 324 100 315 118 315Z" transform="translate(389,0)"></path><path data-c="29" d="M78-245C138-199 188-131 229-40 268 47 288 129 288 208L288 292C288 371 268 453 229 540 188 631 138 699 78 745 75 747 72 748 71 748 62 748 57 743 57 734 57 730 59 726 62 723 114 683 156 617 187 526 214 447 228 369 228 292L228 208C228 131 214 53 187-26 156-117 114-183 62-223 59-227 57-231 57-234 57-243 62-248 71-248 72-248 75-247 78-245Z" transform="translate(889,0)"></path></g></g></g></g></svg></g></g></g></g></svg></mjx-container></p><h3 id="哈代空间（Hardy-space）"><a href="#哈代空间（Hardy-space）" class="headerlink" title="哈代空间（Hardy space）"></a>哈代空间（Hardy space）</h3><p>Hardy 在复分析中研究了与复平面单位圆上的函数有关的一类函数空间，这些空间现在被称为 “哈代空间”，在傅里叶分析和算子理论中具有重要作用。</p><p>在单位圆盘上的哈代空间<mjx-container class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align:0" xmlns="http://www.w3.org/2000/svg" width="3.104ex" height="1.545ex" role="img" focusable="false" viewBox="0 -683 1372.2 683"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="H^p"><g data-mml-node="msup" data-latex="H^p"><g data-mml-node="mi" data-latex="H"><path data-c="1D43B" d="M881 668C881 678 875 683 863 683L736 680 609 683C593 683 586 674 586 659 586 652 589 647 594 646 604 645 612 644 617 644 648 643 665 641 670 640 675 639 678 636 678 631 677 628 676 622 674 613L615 375 321 375 379 602C384 623 393 636 406 641 413 643 430 644 458 644 485 644 496 644 496 668 496 678 490 683 478 683L351 680 223 683C207 683 200 674 200 659 200 652 203 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34-155-5-155-23-155-32-163-32-178-32-189-26-194-15-194 0-194 52-191 67-191 86-191 146-194 165-194 180-194 187-186 187-170 187-160 178-155 159-155 141-155 114-156 114-144 114-131 155 26 159 43 179 6 209-13 249-13 314-13 371 20 421 87 467 148 490 213 490 280 490 367 438 442 355 442M352 412C391 412 411 382 411 323 411 296 405 260 394 215 371 128 342 71 307 42 286 25 267 16 248 16 219 16 199 29 188 56 179 77 174 92 174 100L225 309C230 331 248 354 276 377 304 400 329 412 352 412Z"></path></g></g></g></g></svg></mjx-container> （<mjx-container class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align:-.566ex" xmlns="http://www.w3.org/2000/svg" width="1.138ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 503 1000"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="p \in [0, +\infty] "><g data-mml-node="mi" data-latex="p"><path data-c="1D45D" d="M355 442C311 442 269 419 228 373 217 419 186 442 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367 172 360 128 347 48 310 8 249 8 226 8 203 17 182 34 155 57 139 104 132 176 129 201 128 253 128 332 128 419 131 480 136 513 145 568 163 603 191 618 213 630 232 636 249 636 314 636 350 583 361 524Z"></path></g><g data-mml-node="mo" data-latex="," transform="translate(1722.8,0)"><path data-c="2C" d="M139 106C107 106 86 82 86 50 86 20 109-5 139-5 153-5 165-1 174 8L175 0C175-63 154-117 112-160 105-168 101-174 101-178 101-188 105-193 114-193 123-193 135-181 152-158 186-110 203-57 203 0 203 53 185 106 139 106Z"></path></g></g></g></svg><mjx-break size="2"></mjx-break><svg style="vertical-align:-.566ex" xmlns="http://www.w3.org/2000/svg" width="4.652ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 2056 1000"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="p \in [0, +\infty] "><g data-mml-node="mo" data-latex="+"><path data-c="2B" d="M698 274 413 274 413 559C413 575 405 583 389 583 373 583 365 575 365 559L365 274 80 274C64 274 56 266 56 250 56 234 64 226 80 226L365 226 365-59C365-75 373-83 389-83 405-83 413-75 413-59L413 226 698 226C714 226 722 234 722 250 722 263 711 274 698 274Z"></path></g><g data-mml-node="mi" data-latex="\infty" transform="translate(778,0)"><path data-c="221E" d="M749-11C807-11 855 13 892 60 926 104 943 156 943 216 943 275 926 327 893 371 856 418 809 442 752 442 684 442 625 416 576 364 547 332 524 303 507 278 464 329 435 361 421 373 367 419 310 442 250 442 192 442 144 418 107 371 73 327 56 275 56 215 56 156 73 104 106 60 143 13 190-11 247-11 315-11 374 15 423 67 452 99 475 128 492 153 535 102 564 70 578 58 632 12 689-11 749-11M913 216C913 188 911 168 908 156 903 137 890 117 869 94 840 61 805 44 765 44 722 44 680 67 637 113 592 168 559 209 538 237 601 348 675 403 759 403 852 403 913 314 913 216M86 215C86 260 100 299 128 334 156 369 191 387 234 387 277 387 319 364 362 318 407 263 440 222 461 194 398 83 324 28 240 28 147 28 86 117 86 215Z"></path></g><g data-mml-node="mo" data-latex="]" transform="translate(1778,0)"><path data-c="5D" d="M45-250 164-250 164 750 45 750C30 750 22 742 22 726 22 710 30 702 45 702L119 702 119-202 45-202C30-202 22-210 22-226 22-242 30-250 45-250Z"></path></g></g></g></svg></mjx-container> ）定义为所有满足如下不等式的全纯函数<mjx-container class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align:-.464ex" xmlns="http://www.w3.org/2000/svg" width="1.249ex" height="2.059ex" role="img" focusable="false" viewBox="0 -705 552 910"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="f"><g data-mml-node="mi" data-latex="f"><path data-c="1D453" d="M552 633C552 677 509 705 462 705 400 705 357 665 334 586 329 568 318 517 302 433L237 433C215 433 204 432 204 411 204 400 214 395 235 395L295 395 222 8C211-49 201-91 192-119 180-156 163-175 141-175 126-175 114-171 103-164 135-159 151-140 151-108 151-82 138-69 111-69 77-69 53-99 53-133 53-177 94-205 141-205 166-205 189-195 208-174 240-141 265-94 283-31 294 8 304 46 311 84L369 395 451 395C474 395 484 396 484 419 484 428 474 433 454 433L377 433C383 474 411 625 420 644 430 665 444 675 462 675 477 675 490 671 501 664 470 657 454 639 454 608 454 582 467 569 494 569 528 569 552 599 552 633Z"></path></g></g></g></svg></mjx-container> ：</p><p><mjx-container class="MathJax" jax="SVG" overflow="overflow" display="true" width="full" style="min-width:44.203ex"><svg style="vertical-align:-2.73ex;min-width:44.203ex" xmlns="http://www.w3.org/2000/svg" width="100%" height="6.592ex" role="img" focusable="false"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(0.0181,-0.0181) translate(0, -1706.9)"><g data-mml-node="math" data-latex="
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transform="translate(0,557.2)"><text data-id-align="true"></text><g data-idbox="true" transform="translate(0,-748)"><g data-mml-node="mtext" data-latex="\text{(6)}"><path data-c="28" d="M318-248C327-248 332-243 332-234 332-231 330-227 327-223 275-183 233-117 202-26 175 53 161 131 161 208L161 292C161 369 175 447 202 526 233 617 275 683 327 723 330 726 332 730 332 734 332 743 327 748 318 748 317 748 314 747 311 745 251 699 201 631 160 540 121 453 101 371 101 292L101 208C101 129 121 47 160-40 201-131 251-199 311-245 314-247 317-248 318-248Z"></path><path data-c="36" d="M383 504C416 504 432 521 432 555 432 627 378 666 304 666 221 666 155 627 106 548 63 480 42 403 42 316 42 189 65 100 112 47 152 1 198-22 251-22 312-22 362 1 401 47 438 91 457 144 457 205 457 266 439 318 403 361 365 407 316 431 257 431 205 431 165 402 138 346L138 352C138 465 166 561 226 605 252 624 279 633 306 633 342 633 368 623 385 602 351 602 334 583 334 553 334 525 355 504 383 504M344 340C355 317 361 272 361 206 361 141 356 98 345 76 325 35 294 14 251 14 222 14 200 24 184 44 171 60 162 74 158 85 146 116 140 163 140 227 140 255 144 282 151 308 164 355 201 399 256 399 295 399 325 379 344 340Z" transform="translate(389,0)"></path><path data-c="29" d="M78-245C138-199 188-131 229-40 268 47 288 129 288 208L288 292C288 371 268 453 229 540 188 631 138 699 78 745 75 747 72 748 71 748 62 748 57 743 57 734 57 730 59 726 62 723 114 683 156 617 187 526 214 447 228 369 228 292L228 208C228 131 214 53 187-26 156-117 114-183 62-223 59-227 57-231 57-234 57-243 62-248 71-248 72-248 75-247 78-245Z" transform="translate(889,0)"></path></g></g></g></g></svg></g></g></g></g></svg></mjx-container></p><h3 id="哈代-温伯格原理"><a href="#哈代-温伯格原理" class="headerlink" title="哈代-温伯格原理"></a>哈代 - 温伯格原理</h3><p>Hardy 对遗传学也有贡献，著名的哈代 - 温伯格原理（Hardy-Weinberg principle），就是 Hardy 的板球友向他请教孟德尔遗传理论的结果。哈代 - 温伯格原理指出，在没有其他进化影响的情况下，一个群体中的等位基因频率和基因型频率将从一代到下一代保持不变。这些影响包括遗传漂变、配偶选择、正向交配、自然选择、性选择、突变、基因流动、减数驱动、遗传搭车效应、群体瓶颈效应、奠基者效应、近亲繁殖和远亲繁殖衰退。</p></div></div><div class="footer"></div><div class="article-meta" id="bottom"><div class="new-meta-box"></div></div><div class="prev-next"><a class="prev" href="/notes/person/11"><p class="title"><i class="fa-solid fa-chevron-left" aria-hidden="true"></i>Ramanujan印度数学奇才的传奇一生：拉马努金的贡献与影响</p><p class="content">本文记述了印度数学天才拉马努金的生平与数学贡献。他未受正规教育却独立发现大量深刻公式，后赴剑桥与哈代合作，在数论、模形式和划分函数等领域取得奠基性成果，其猜想深远影响了现代数学发展</p></a><a class="next" href="/notes/person/13"><p class="title">Littlewood英国数学家李特尔伍德生平与数学贡献<i class="fa-solid fa-chevron-right" aria-hidden="true"></i></p><p class="content">约翰·埃德森·李特尔伍德是英国数学家，以其与哈代的长期合作闻名。他们在解析数论、函数论等领域取得开创性成果，如关于素数定理误差项的研究及著名的Hardy-Littlewood猜想。他一生深受抑郁症困扰，但学术成就卓著</p></a></div><div class="recommended-article"><div class="recommended-article-header"><i class="fa-solid fa-bookmark fa-fw" aria-hidden="true"></i> <span>推荐阅读</span></div><div 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fa-comments"></i> 留言区</p><div id="layoutHelper-comments"></div></article></div><aside id="l_side" itemscope="" itemtype="http://schema.org/WPSideBar"><section class="widget text desktop mobile pjax"><header><a href="/notes/"><i class="fa-duotone fa-book fa-fw" aria-hidden="true"></i> <span class="name">Notes</span></a></header><div class="content"></div></section><section class="widget list group desktop mobile pjax"><header><i class="fa-duotone fa-p fa-fw" aria-hidden="true"></i> <span class="name">History of mathematics</span></header><div class="content"><ul class="list entry navigation"><li><a class="flat-box" title="/notes/person/" href="/notes/person/" active-action="action-notesperson"><div class="name"> Welcome</div></a></li><li><a class="flat-box" title="/notes/person/1" href="/notes/person/1" active-action="action-notesperson1"><div class="name"> Hilbert</div></a></li><li><a class="flat-box" title="/notes/person/2" href="/notes/person/2" active-action="action-notesperson2"><div class="name"> Thompson</div></a></li><li><a class="flat-box" title="/notes/person/3" href="/notes/person/3" active-action="action-notesperson3"><div class="name"> Conway</div></a></li><li><a class="flat-box" title="/notes/person/4" href="/notes/person/4" active-action="action-notesperson4"><div class="name"> Borcherds</div></a></li><li><a class="flat-box" title="/notes/person/5" href="/notes/person/5" active-action="action-notesperson5"><div class="name"> Wall</div></a></li><li><a class="flat-box" title="/notes/person/6" href="/notes/person/6" active-action="action-notesperson6"><div class="name"> Waldhausen</div></a></li><li><a class="flat-box" title="/notes/person/7" href="/notes/person/7" active-action="action-notesperson7"><div class="name"> Dehn</div></a></li><li><a class="flat-box" title="/notes/person/8" href="/notes/person/8" active-action="action-notesperson8"><div class="name"> Whitehead</div></a></li><li><a class="flat-box" title="/notes/person/9" href="/notes/person/9" active-action="action-notesperson9"><div class="name"> Iwaniec</div></a></li><li><a class="flat-box" title="/notes/person/10" href="/notes/person/10" active-action="action-notesperson10"><div class="name"> Sarnak</div></a></li><li><a class="flat-box" title="/notes/person/11" href="/notes/person/11" active-action="action-notesperson11"><div class="name"> Ramanujan</div></a></li><li><a class="flat-box" title="/notes/person/12" href="/notes/person/12" active-action="action-notesperson12"><div class="name"> Hardy</div></a></li><li><a class="flat-box" title="/notes/person/13" href="/notes/person/13" active-action="action-notesperson13"><div class="name"> Littlewood</div></a></li><li><a class="flat-box" title="/notes/person/14" href="/notes/person/14" active-action="action-notesperson14"><div class="name"> von Neumann</div></a></li><li><a class="flat-box" title="/notes/person/15" href="/notes/person/15" active-action="action-notesperson15"><div class="name"> Irving Segal</div></a></li><li><a class="flat-box" title="/notes/person/16" href="/notes/person/16" active-action="action-notesperson16"><div class="name"> Birkhoff</div></a></li><li><a class="flat-box" title="/notes/person/17" href="/notes/person/17" active-action="action-notesperson17"><div class="name"> Wiener</div></a></li><li><a class="flat-box" title="/notes/person/18" href="/notes/person/18" active-action="action-notesperson18"><div class="name"> Shannon</div></a></li><li><a class="flat-box" title="/notes/person/19" href="/notes/person/19" active-action="action-notesperson19"><div class="name"> Hedlund</div></a></li><li><a class="flat-box" title="/notes/person/20" href="/notes/person/20" active-action="action-notesperson20"><div class="name"> Gibbs</div></a></li><li><a class="flat-box" title="/notes/person/21" href="/notes/person/21" active-action="action-notesperson21"><div class="name"> Veblen</div></a></li><li><a class="flat-box" title="/notes/person/22" 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