哈拉尔·玻尔Harald Bohr

哈拉尔·玻尔是尼尔斯·玻尔的弟弟。他们的父亲Christian Bohr是哥本哈根大学生理学教授。Christian Bohr以其关于呼吸的物理和化学方面的研究而闻名。玻尔和尼尔斯·玻尔的母亲埃伦·阿德勒玻尔出身于一个富裕的犹太家庭，家族成员在丹麦的银行业和政治中举足轻重。

玻尔在哥本哈根大学学习数学。他于1904年进入该大学，并很快成为丹麦知名人物，不是因为他的数学，而是因为他的足球技能。他入选丹麦足球队，该队在1908年伦敦奥运会上获得第二名。当他的学位论文在哥本哈根大学接受审查时，希望参加这场公开答辩的足球迷比数学家还多！

数学很快对玻尔来说变得比足球更重要，他于1915年成为哥本哈根理工学院的数学教授。然后，在1930年，他被任命为哥本哈根大学数学教授。尽管他从未完全达到他哥哥尼尔斯·玻尔的名望（除了作为足球运动员！），他确实做出了一些极其重要的数学工作。也许令人惊讶的是，玻尔和尼尔斯·玻尔没有更频繁地合作。他们只发表了一篇联合论文。

玻尔研究约翰·彼得·古斯塔夫·勒热纳·狄利克雷级数，并将分析应用于数论。他与当时在哥廷根的埃德蒙·朗道合作，研究波恩哈德·黎曼ζ函数。1914年，他们证明了关于zeta函数零点分布的玻尔-埃德蒙·朗道定理。

关于zeta函数的这些重要工作中，有些是玻尔独自完成的，有些则来自与埃德蒙·朗道的合作。在他们证明的众多引人注目的结果中，最令人印象深刻的是一些迈向黎曼假设证明的重大步骤（然而，该猜想至今仍未获证）。玻尔和埃德蒙·朗道证明了zeta函数的零点除无穷小比例外全部位于直线$s = \large\frac{1}{2}\normalsize$的某个小邻域内。

玻尔对哪些函数可由约翰·彼得·古斯塔夫·勒热纳·狄利克雷级数表示的兴趣，促使他创立了概周期函数理论。他在1923年至1926年间创立了这一理论，正是这项工作使他的名字如今与之最为紧密地联系在一起。粗略地说，概周期函数是指这样一个函数：经过一个周期后，其取值与前一周期中的取值相差不超过e。玻尔于1924年至1926年间在Acta Mathematica上发表了关于这一主题的三部重要著作。

概周期函数的基本定理是Fourier series的马克-安托万·帕塞瓦尔恒等式的推广。这一结果引导玻尔得到了关于用指数函数对概周期函数进行一致逼近的结果。

爱德华·查尔斯·蒂奇马什 在 [10] 中总结了他关于几乎周期函数的工作：-

一般理论是为实变量的情形发展起来的，然后，在此光照下，发展了复变量几乎周期函数的最美丽的理论。……实变量几乎周期函数理论的创立是一项非凡力量的成就，但并非基于最先进的方法，主要结果很快就被简化和改进。然而，复变量几乎周期函数的理论至今仍保持着玻尔所赋予的完美形式。

在建立了几乎周期函数理论之后，玻尔的数学工作完全致力于推进这一主题。他继续工作直到去世前不久，事实上，他在去世前四个月还参加了在马萨诸塞州剑桥举行的国际数学家大会。

阿布拉姆·萨莫伊洛维奇·贝西科维奇在[3]中写道：-

玻尔一生中大部分时间都是一个病人。他常常遭受严重头痛的折磨，不得不避免一切脑力劳动。作为一个人，玻尔不亚于作为数学家的玻尔那样杰出。他是一个智力精妙的人，在许多方面和谐发展。他也是一个极其仁慈的人。他对学生、同事和朋友，以及学术界的难民的帮助确实慷慨。一旦他决定帮助，他就无所顾忌，而且很少失败。他对文学非常敏感。他最喜欢的作家是狄更斯；他深深钦佩狄更斯对人类的热爱，并深深欣赏他的幽默。

Harald Bohr was a younger brother of Niels Bohr. Their father, Christian Bohr, was professor of physiology at the University of Copenhagen. Christian Bohr was famous for his work on the physical and chemical aspects of respiration. Harald and Niels Bohr's mother, Ellen Adler Bohr, came from a wealthy Jewish family with family members who were important in banking and in politics in Denmark.

Harald studied mathematics at the University of Copenhagen. He entered the University in 1904 and quickly became a well known Danish personality, not for his mathematics but rather for his soccer skills. He was in the Danish soccer team which was placed second in the 1908 Olympic games in London. When his doctoral dissertation was examined at the University of Copenhagen, there were more soccer fans wishing to attend this public examination than there were mathematicians!

Mathematics soon became more important to Bohr than soccer and he became professor of mathematics in the Polytechnic Institute in Copenhagen in 1915. Then, in 1930, he was appointed professor of mathematics at the University of Copenhagen. Although he never quite attained the fame of his brother Niels (except as a soccer player!), he did produce some mathematics of the very highest importance. It is perhaps surprising that Harald and Niels did not collaborate more frequently. They only published one joint paper.

Harald Bohr worked on Dirichlet series, and applied analysis to the theory of numbers. He collaborated with Edmund Landau, who was at this time at Göttingen, in studying the Riemann zeta function. In 1914 they proved the Bohr-Landau theorem on the distribution of zeros of the zeta function.

Some of this important work on the zeta function was due to Bohr alone, some came from the collaboration with Landau. Some of the most impressive from the many striking results which they proved were major steps towards a proof of the Riemann hypothesis (which, however, is still unproved). Bohr and Landau proved that all but an infinitesimal proportion of the zeros of the zeta function lie in a small neighbourhood of the line $s = \large\frac{1}{2}\normalsize$ .

Bohr's interest in which functions could be represented by a Dirichlet series led him to devise the theory of almost periodic functions. He founded this theory between the years 1923 and 1926 and it is with this work that his name is now most closely associated. Roughly speaking an almost periodic function is one which, after a period, takes values within e of its values in the previous period. Bohr published three major works on this topic in Acta Mathematica between 1924 and 1926.

The fundamental theorem for almost periodic functions is a generalisation of the Parseval identity for Fourier series. This result lead Bohr to a result on the uniform approximation to almost periodic functions by exponential functions.

Titchmarsh, writing in [10], sums up his work on almost periodic functions:-

The general theory was developed for the case of a real variable, and then, in the light of it, was developed the most beautiful theory of almost periodic functions of a complex variable. ... The creation of the theory of almost periodic functions of a real variable was a performance of extraordinary power, but was not based on the most up-to-date methods, and the main results were soon simplified and improved. However, the theory of almost periodic functions of a complex variable remains up to now in the same perfect form in which it was given by Bohr.

After setting up the theory of almost periodic functions, Bohr's mathematical work became devoted exclusively to furthering the subject. He continued his work until shortly before his death, in fact he attended the International Congress of Mathematicians in Cambridge, Massachusetts four months before his death.

Besicovitch writes in [3]:-

For most of his life Bohr was a sick man. He used to suffer from bad headaches and had to avoid all mental effort. Bohr the man was not less remarkable than Bohr the mathematician. He was a man of refined intellect, harmoniously developed in many directions. He was also a most humane person. His help to his pupils, to his colleagues and friends, and to refugees belonging to the academic world was generous indeed. Once he had decided to help he stopped at nothing and he seldom failed. He was very sensitive to literature. His favourite author was Dickens; he had a deep admiration of Dickens' love of the human being and deep appreciation of his humour.