罗尔夫·内万林纳Rolf Nevanlinna

罗尔夫·内万林纳的父母是奥托·威廉·内奥维乌斯和玛格丽特·龙贝里，他们说瑞典语和芬兰语。当然，这立刻引出了一个问题：既然他父亲姓内奥维乌斯，内万林纳这个名字从何而来。事实上，内万林纳在生命的最初十年里被称为内万林纳·内奥维乌斯，直到1906年他父亲奥托更改了家族姓氏后才成为内万林纳。奥托生于1867年，来自一个数学世家，在赫尔辛基大学学习数学、物理学和天文学。获得物理学博士学位后，他在普尔科沃天文台跟随天文学家赫尔曼·龙贝里学习，并于1892年与龙贝里的女儿玛格丽特结婚。他们定居在约恩苏，奥托在那里教授物理学，他们的四个孩子也出生在那里：弗里蒂奥夫（生于1894年）、内万林纳（本传记的主人公，生于1895年）、安娜（生于1896年）和埃里克（生于1901年）。弗里蒂奥夫也成为了一名数学家，他与内万林纳合作发表了一些出版物，我们将在下文提及。

1902年，七岁的内万林纳在约恩苏开始上学。由于已在家中学会读写，他直接进入二年级，但仍觉得功课无聊，过了一段时间便拒绝去学校。他在家接受了大约十八个月的教育，后来父亲被任命为教师，全家迁往赫尔辛基，他便开始就读于父亲当时教授数学和物理的那所学校。在学校里，他在已掌握的语言之外又学习了法语和德语。这个家庭还热爱音乐，这得益于母亲的鼓励[12]：-

玛加蕾特是一位出色的钢琴家，弗里蒂奥夫和内万林纳会躺在钢琴下听她演奏。13岁时，他们进入管弦乐学校，成为有造诣的音乐家——弗里蒂奥夫演奏大提琴，内万林纳演奏小提琴。通过管弦乐学校的免费票，他们得以了解并热爱伟大作曲家的音乐，如巴赫、贝多芬、勃拉姆斯、舒伯特、舒曼、肖邦和李斯特，以及由作曲家本人指挥的西贝柳斯（1865-1957）的早期交响曲。内万林纳于1907年首次接触西贝柳斯的音乐，当时他听到了他的第三交响曲。尽管后来他结识了大卫·希尔伯特、阿尔伯特·爱因斯坦、托马斯·曼和其他名人，内万林纳说没有人像西贝柳斯那样对他产生如此强烈的影响。兄弟俩与母亲一起演奏三重奏，他们对音乐的热爱——尤其是对室内乐的热爱——持续了一生。

在赫尔辛基高中，内万林纳的兴趣首先是古典学，其次是数学。他受教于多位优秀教师，但也许其中最好的是他自己的父亲，在学校的最后几年教他物理和数学。1913年，他以优异的成绩毕业，尽管他不是那一届最好的学生。然而，在1913年夏天，他在开始大学学习之前阅读了恩斯特·列奥纳德·林德勒夫的 Introduction to Higher Analysis，从而超越了学校教学大纲。从那时起，他成为一名热情的分析学家，毕生致力于这一领域。（恩斯特·列奥纳德·林德勒夫是内万林纳父亲的表亲，因此也是这个数学家族的一员。）

内万林纳于1913年进入赫尔辛基大学。在那里，他受到恩斯特·列奥纳德·林德勒夫教学的启发，但次年第一次世界大战的爆发威胁到他的学业。芬兰在内万林纳的一生中都处于俄罗斯的统治之下，但这种统治在20世纪最初几年有所加强，导致国内出现了强烈的独立运动。战争开始后，独立运动向德国寻求支持，年轻志愿者前往德国接受军事训练并组建了猎兵营。内万林纳渴望去德国加入猎兵营，但父母说服他应该继续学业。由于他身体相当虚弱，他避免了被征入俄罗斯军队，因此能够在整个战争年代继续学业。事实上，1917年12月芬兰宣布从俄罗斯独立后，赫尔辛基成为内战的现场。他的学位论文Über beschränkte Funktionen die in gegebenen Punkten vorgeschriebene Werte annehmenⓉ（《在给定点规定值的有限函数》），于1919年提交，研究了满足某些固定条件的正则函数的存在性。这项工作展现了他所有工作中典型的细致和对细节的关注。他于1919年6月2日获得博士学位，并在同一天与表妹玛丽结婚。他们订婚已有两年，但自1911年内万林纳与哥哥弗里蒂奥夫一起回到约恩苏回忆童年时，他们就已是朋友。内万林纳和玛丽有四个孩子：凯（生于1920年）、哈里（生于1922年）、阿尔内（生于1925年）和西尔维（生于1930年）。

1919年芬兰没有大学职位，因此内万林纳成为一名学校教师。他的兄弟弗里蒂奥夫在保险公司萨拉马担任数学家，除了全职学校教学外，内万林纳也为萨拉马工作。1920年，他收到埃德蒙·朗道的邀请前往哥廷根，但他没有立即接受。1922年，他成为赫尔辛基大学的讲师，但直到1926年被任命为赫尔辛基的教授才放弃学校教学。值得注意的是，尽管工作量如此繁重，正是在这些年里，他产生了今天所谓的“内万林纳理论”。他的工作习惯或许可以解释如此高产的工作成果。这些在[12]中描述如下：-

内万林纳从不需要太多睡眠。20世纪30年代末，他常常4点起床，10点与家人相聚直到午餐时间，然后再工作到7点，晚上与家人共度。

Olli Lehto是内万林纳的学生，在[20]中描述了内万林纳理论的诞生。沃尔特·库尔特·海曼在一篇评论中总结了Lehto的论文，该论文：-

……极其有趣地叙述了值分布理论的起源。他从卡尔·魏尔斯特拉斯构造具有预先给定零点序列的整函数开始，接着是埃德蒙·拉盖尔引入亏格，以及儒勒·昂利·庞加莱和雅克·阿达马关于零点的收敛指数与函数的级之间联系的结果，后者最初由埃米尔·博雷尔定义。埃米尔·博雷尔还证明了埃米尔·皮卡定理的推广，该定理指出任何整函数的a-点的收敛指数等于其级，至多有一个a值例外。这就是内万林纳引入其特征函数之前的情况，特征函数不仅将理论推广到亚纯函数，而且即使对整函数的特殊情形，也通过他的第一和第二基本定理以及亏量关系，得到了更为精确的结果表述。这一发展在20世纪20年代通过R 内万林纳的一系列论文推进，最终体现在专著《Le Théorème de Picard-Borel et la théorie des fonctions méromorphes》Ⓣ（皮卡-博雷尔定理与亚纯函数理论）（1929年）中。

内万林纳理论的主要结果出现在1925年那篇100页的论文Zur Theorie der meromorphen Funktionen Ⓣ（论亚纯函数理论）中。这篇论文被赫尔曼·外尔描述为20世纪少数几个伟大的数学事件之一。为了说明内万林纳惊人的产出，我们注意到这篇100页的论文只是他1925年发表的八篇论文之一，而在此前三年中他已发表了十三篇论文。当我们意识到这一巨大产出是在他同时担任中学教师和大学讲师的那些年里取得的时，就更令人惊叹了。

内万林纳余生一直在赫尔辛基大学任职，除了他前往许多不同国家的广泛旅行。也许他对结识其他数学家的热情是由他参加1922年在赫尔辛基举行的第五届斯堪的纳维亚数学大会所激发的。在那里他第一次遇到了约斯塔·米塔格-莱弗勒——后者已在赫尔辛基担任教授五年，远在内万林纳出生之前。他还遇到了哈拉尔·玻尔、尼尔斯·玻尔、Torsten Carleman和哈拉尔德·克拉梅尔等人。他第一次出国是1924年，为迟答埃德蒙·朗道的邀请而前往哥廷根。在哥廷根，除了兰道，他还遇到了大卫·希尔伯特、理查·科朗特和埃米·诺特；在德国期间，他还在慕尼黑遇到了帕维尔·亚历山德罗夫、帕维尔·萨穆伊洛维奇·乌雷松和康斯坦丁·卡拉西奥多里。后来的访问包括1926年去巴黎，在那里他遇到了雅克·阿达马、保罗·蒙泰尔，还去精神病院探望了André Bloch。1928年访问苏黎世时，他遇到了A. Speiser，并对波恩哈德·黎曼曲面的分类问题产生了兴趣。两年后，内万林纳离开苏黎世去哥廷根接任讲席后，赫尔曼·外尔的讲席提供给了内万林纳，但内万林纳拒绝了。正是在20世纪20年代后半期，拉尔斯·阿尔福斯跟随内万林纳学习，于1930年获得博士学位，并陪同他的学位论文导师前往苏黎世。拉尔斯·阿尔福斯对阿尔诺·当茹瓦猜想的证明——这使他在1936年获得了菲尔兹奖——遵循了内万林纳在苏黎世时提出的建议。这些几何思想标志着内万林纳研究中的一个新元素，在他20世纪30年代初发表的论文中显而易见。这些思想在他1936年的著作Eindeutige analytische Funktionen Ⓣ（唯一解析函数）中有详细阐述，该书讨论了他发明调和测度相关的思想。沃尔特·库尔特·海曼写道，这本书[12]：-

……自写成以来一直激励着函数论学者，并且像大多数后来的书一样，已被翻译成俄文和英文，还有几个德文版本。

他继续在赫尔辛基工作，但也继续热爱研究访问，1936-37年作为访问教授在哥廷根度过。回到芬兰后，内万林纳和他的妻子玛丽于1937年在赫尔辛基以西约45公里的洛赫亚购买了一处避暑别墅，这里成为许多访问数学家在夏季度过时光的地方。他于1941年成为赫尔辛基大学校长，并在第二次世界大战期间一直担任此职。战争期间，他通过改进弹道表为战争努力做出贡献，并被授予自由十字勋章。

从学生时代起，当他希望德国帮助芬兰摆脱俄罗斯统治时，内万林纳就以与德国有联系而闻名。1945年德国战败后，赫尔辛基大学当局认为，让一个已知有德国同情心的人担任校长是不合适的，因此他被要求辞去该职务。然而，他保留了在赫尔辛基的讲席，但在1946年10月，他再次访问了苏黎世。这标志着他作为客座讲师在苏黎世为期十五年的时期的开始，这一时期在[7]中有详细描述。内万林纳在苏黎世和赫尔辛基讲授的课程构成了他的著作Uniformisierung Ⓣ(Uniformity)（1953年）的基础。M Heins以以下段落开始对该书进行详细评论：-

自从赫尔曼·外尔著名的《Idee der Riemannschen Fläche》Ⓣ（《波恩哈德·黎曼曲面的思想》）（1913年）问世以来，抽象波恩哈德·黎曼曲面理论经历了相当大的发展，尤其是在过去二十五年中。尽管如此，除了Stoilow的专著外，迄今为止以书籍形式出现的关于这一主题的著作很少。内万林纳的论著基于他在苏黎世和赫尔辛基的讲座，是关于这一主题的新一代书籍中的第一部。作者的目标是提供抽象波恩哈德·黎曼曲面理论的介绍，其中单值化理论是关注的焦点。

第二次世界大战后，在与苏黎世联系期间，他的数学兴趣转向变分法及其在物理学中的应用。例如，他写了论文Calculus of variation and partial differential equations（1967年）。

内万林纳一生都积极参与音乐活动。1945年，他正在组建一个室内乐协会时，遇到了作家兼翻译家Sinikka Kallio-Visapaa。他的第一次婚姻解除后，于1958年与Sinikka结婚。他们有一个女儿Kristiina。

我们已经提到了内万林纳和他的兄弟Frithiof之间的数学合作。沃尔特·库尔特·海曼在[12]中写到了他们在20世纪20年代的合作：-

晚上和星期天是研究的绝佳时间，内万林纳理论深受内万林纳与Frithiof讨论的影响，这些讨论持续了他们一生。他们会沿着一个大广场的两侧走来走去，谈论数学。

他们一起撰写论文，例如：Über die Eigenschaften analytischer Funktionen in der Umgebung einer singulären Stelle oder Linie Ⓣ（《关于解析函数在奇点或奇线附近的性质》）（1922年），Über die Nullstellen der Riemannschen Zetafunktion Ⓣ（《关于黎曼 zeta函数的零点》）（1924年），以及Über die Nullstellen der Riemannschen Zetafunktion Ⓣ（《关于黎曼 zeta函数的零点》）（1925年）。多年后，他们还合作撰写了Absolute analysis一书（1959年）。英译本于1973年出版。内万林纳与V Paatero合著了Einführung in die Funktionentheorie Ⓣ（《函数论导论》）（1965年）。A J Lohwater在书评开头写道：-

这本书精彩地讲述了赫尔辛基大学传统的复变函数入门课程，其前七章反映了恩斯特·列奥纳德·林德勒夫入门教材的影响。

除了教科书和专著外，内万林纳还出版了两本书和五十多篇文章，旨在向非数学家介绍数学思想。其中一个例子是他的书Raum, Zeit und Relativität Ⓣ（《空间、时间与相对论》）（1964年）。他关于数学教学的思想的一个例子，这引起了他相当大的兴趣，见于他的论文Reform in teaching mathematics（1966年）。

内万林纳因其贡献获得了许多荣誉。他被授予国际Wihuri科学家与艺术家奖（1958年）和Henrik Steffens北欧文化奖（1967年）。他被选为德意志科学院荣誉成员（1938年）；Finnish Mathematical Society（1955年）；London Mathematical Society（1959年）；Swiss Mathematical Society（1962年）；芬兰精算师学会（1965年）；数学与物理教师协会（1965年）；Göttingen Academy of Sciences（1967年）；Royal Swedish Academy of Sciences（1967年）；Danish Academy of Sciences（1967年）；German Academy of Sciences Leopoldina（1967年）；Hungarian Academy of Sciences（1970年）；法兰西学院（1967年）；以及Finnish Academy of Science and Letters（1975年）。他被海德堡大学（1936年）、布加勒斯特大学（1942年）、吉森大学（1952年）、柏林大学（1955年）、于韦斯屈莱大学（1969年）、格拉斯哥大学（1969年）、乌普萨拉大学（1974年）和伊斯坦布尔大学（1976年）授予荣誉学位。他于1959年至1962年担任国际数学联盟主席。

内万林纳一直发表论文，直到1980年因癌症去世。自1982年起，在国际数学家大会上颁发一个奖项——内万林纳奖。每届大会颁发一个奖项，授予从事信息科学数学方面研究的年轻数学家。

Rolf Nevanlinna's parents were Otto Wilhelm Neovius and Margarete Romberg who spoke both Swedish and Finnish. Of course this immediately prompts the question of where the name Nevanlinna came from since his father's name was Neovius. In fact Rolf Nevanlinna was known as Rolf Neovius for the first ten years of his life, only becoming Nevanlinna in 1906 when his father Otto changed the family name. Otto, born in 1867, came from a mathematical family and studied mathematics, physics and astronomy at Helsinki University. After obtaining a doctorate in physics he studied with the astronomer Herman Romberg at Pulkova Observatory and married Romberg's daughter Margarete in 1892. They settled in Joensuu where Otto taught physics, and there their four children were born: Frithiof (born 1894), Rolf (the subject of this biography, born 1895), Anna (born 1896) and Erik (born 1901). Frithiof also became a mathematician and he collaborated with Rolf on a number of publications which we mention below.

In 1902, at the age of seven, Rolf began his schooling in Joensuu. Having already been taught to read and write at home, he went straight into the second grade but he still found the work boring and after a while he refused to attend the school. He was educated at home for about eighteen months but then, after the family moved to Helsinki when he father was appointed as a teacher, he began attending the school there where his father was now teaching mathematics and physics. At school he learnt French and German to add to the languages he already spoke. The family also had a love for music encouraged by their mother [12]:-

Margarete was an excellent pianist and Frithiof and Rolf would lie under the piano and listen to her playing. At 13 they went to orchestra school and became accomplished musicians - Frithiof on the 'cello and Rolf on the violin. Through free tickets from the orchestra school they got to know and love the music of the great composers, Bach, Beethoven, Brahms, Schubert, Schumann, Chopin and Liszt, as well as the early symphonies of Sibelius (1865-1957), conducted by the composer. Rolf first met Sibelius's music in 1907, when he heard his Third Symphony. Although later he met Hilbert, Einstein, Thomas Mann and other famous people, Rolf said that none had had such a strong effect on him as Sibelius. The boys played trios with their mother and their love of music - in particular of chamber music - lasted all their lives.

At the Helsinki High School Nevanlinna's interests were firstly classics, secondly mathematics. He was taught by a number of excellent teachers, but perhaps the best of them all was his own father who taught him physics and mathematics in his final years at school. He graduated from the school in 1913 having performed very well although he was not the best student in his year. However he went beyond the school syllabus in the summer of 1913 when he read Lindelöf's Introduction to Higher Analysis before beginning his university studies. From that time on he became an enthusiastic analyst working on that topic for the whole of his life. (Ernst Lindelöf was a cousin of Nevanlinna's father and so part of this mathematical family.)

Nevanlinna entered Helsinki University in 1913. There he was inspired by Lindelöf's teaching but the beginning of World War I in the following year threatened to disrupt his studies. Finland had been under Russian domination throughout Nevanlinna's life but this domination had increased during the first years of the 20th century resulting in a strong independence movement within the country. After the war began, the independence movement looked to Germany for support, and young volunteers went to Germany where they received military training and formed the Jägar Battalion. Nevanlinna was keen to go to Germany and join the Jägar Battalion but his parents persuaded him that he should continue with his studies. He avoided being conscripted into the Russian military since he was a rather frail young man and so was able to continue his studies throughout the war years. In fact Helsinki was the scene of a civil war after Finland declared independence from Russia in December 1917. His thesis Über beschränkte Funktionen die in gegebenen Punkten vorgeschriebene Werte annehmen Ⓣ, presented in 1919, investigates the existence of regular functions satisfying certain fixed conditions. The work exhibits the care and attention to detail which was to typify all his work. He was awarded his doctorate on 2 June 1919 and on the same day married his cousin Mary. They had been engaged to marry for two years but they had been friends since 1911 when Nevanlinna, together with his older brother Frithiof, had returned to Joensuu to remember their childhood there. Rolf and Mary had four children: Kai (born 1920), Harri (born 1922), Aarne (born 1925), and Sylvi (born 1930).

University posts were not available in Finland in 1919 so Nevanlinna became a school teacher. His brother Frithiof worked as a mathematician for the insurance company Salama and, in addition to full time school teaching, Rolf Nevanlinna also worked for Salama. In 1920 he received an invitation from Edmund Landau to go to Göttingen but he did not accept immediately. He became a lecturer at Helsinki University in 1922 but he did not give up school teaching until he was appointed professor at Helsinki in 1926. Remarkably, despite this heavy workload, it was during these years that he produced what today is called 'Nevanlinna theory'. His working habits may explain this very high output of work. These are described in [12] as follows:-

Rolf never needed much sleep. In the late 1930's he used to rise at 4, join his family at 10 till lunch time, then work again till 7, and spend the evening with his family.

Olli Lehto, who was a student of Nevanlinna, describes the birth of Nevanlinna theory in [20]. Walter Hayman, in a review, gives a summary of Lehto's paper which:-

... gives a most interesting account of the beginnings of value distribution theory. He starts with the construction by Weierstrass of an entire function with a preassigned sequence of zeros, the introduction by Laguerre of the genus and the results by Poincaré and Hadamard on the link between the exponent of convergence of the zeros and the order of the function which had first been defined by Borel. Borel also proved the extension of Picard's theorem which states that the exponent of convergence of the a-points of any entire function is equal to its order except for at most one value of a. This was the situation before Nevanlinna introduced his characteristic which not only extended the theory to meromorphic functions but also obtained a much more precise formulation of the results even in the special case of entire functions by means of his first and second fundamental theorems and the deficiency relation. The development proceeded in the 1920's ... through R Nevanlinna's successive papers culminating in the monograph 'Le Théorème de Picard-Borel et la théorie des fonctions méromorphes' Ⓣ(1929).

The main results of Nevanlinna theory appeared in the 100 page paper Zur Theorie der meromorphen Funktionen Ⓣ in 1925. This paper was described by Weyl as one of the few great mathematical events of the twentieth century. To illustrate Nevanlinna's remarkable output, we note that this 100 page paper was one of eight papers he published in 1925, with thirteen papers having been published in the preceding three years. It is even more amazing when we realise that this huge output was achieved during years when he was working both as a school teacher and as a university lecturer.

Nevanlinna remained on the staff at the University of Helsinki for the rest of his life except for extensive travels he made to many different countries. Perhaps his enthusiasm for meeting other mathematicians was fired by his attendance at the 5th Scandinavian Mathematical Congress at Helsinki in 1922. There he met Mittag-Leffler for the first time - he had been professor in Helsinki for five years, long before Nevanlinna was born. He also met, among others, Harald Bohr, Niels Bohr, Torsten Carleman and Harald Cramér. His first visit abroad was to belatedly accept Edmund Landau's invitation to go to Göttingen which he did in 1924. In addition to Landau he met Hilbert, Courant and Emmy Noether in Göttingen and, while in Germany he also met Pavel Sergeevich Aleksandrov, Pavel Samuilovich Urysohn, and Constantin Carathéodory in Munich. Later visits included one to Paris in 1926 where he met Hadamard, Montel and he also visited André Bloch in a mental hospital. He met A Speiser in Zürich when he visited there in 1928 and became interested in classification problems of Riemann surfaces. Two years later Nevanlinna was offered Weyl's chair after he had left Zürich to take up the chair at Göttingen, but Nevanlinna refused the chair. It was through the second half of the 1920s that Lars Ahlfors studied with Nevanlinna, obtaining his doctorate in 1930, and he had accompanied his thesis advisor on the trip to Zürich. Ahlfors's proof of the Denjoy Conjecture, which led to him being awarded a Fields Medal in 1936, followed suggestions made by Nevanlinna while in Zürich. The geometric ideas marked a new element in Nevanlinna's research and is evident in papers he published in the early 1930s. They are set out in detail in his the book Eindeutige analytische Funktionen Ⓣ (1936) which discusses ideas related to his invention of harmonic measure. Hayman writes that this book [12]:-

... has inspired function theorists ever since it was written and which, like most of the later books, has been translated into Russian and English as well as having several German editions.

He continued to work at Helsinki, but also continued his love for research visits, spending 1936-37 as a visiting professor at Göttingen. Returning to Finland, Nevanlinna and his wife Mary bought a summer home in Lohja, about 45 km west of Helsinki, in 1937 and this became a place where many visiting mathematicians spent time during the summers. He became rector of the University of Helsinki in 1941, retaining this position for the duration of World War II. During the war he contributed to the war effort by improving ballistic tables and was awarded the Cross of Liberty.

From his student days when he had wanted German help to free Finland of Russian domination, Nevanlinna was known to have German links. After the defeat of Germany in 1945, the authorities at Helsinki University felt that it would be inappropriate to have a rector who was known to have German sympathies so he was asked to resign from that role. He retained his chair at Helsinki, however, but in October 1946 he again visited Zürich. This marks the beginning of a fifteen year period during which he was a guest lecturer in Zürich, a period which is described in detail in [7]. Lecture courses which Nevanlinna gave in Zürich and in Helsinki formed the basis for his book Uniformisierung Ⓣ (1953). M Heins begins a detailed review of the book with the following paragraph:-

Since the appearance of Weyl's celebrated 'Idee der Riemannschen Fläche' Ⓣ (1913) the theory of abstract Riemann surfaces has experienced a considerable development, especially in the last quarter of a century. In spite of this fact, apart from the monograph of Stoilow, very little has appeared on this subject in book form up to the present. The treatise of Nevanlinna, which is based upon his lectures at Zürich and Helsinki, is the first of a new generation of books to appear on the subject. The objective of the author is to provide an introduction to the theory of abstract Riemann surfaces with uniformization theory being the focal point of interest.

After World War II, during his association with Zürich, his mathematical interests moved to the calculus of variations and applications to physics. For example he wrote the paper Calculus of variation and partial differential equations (1967).

Throughout his life Nevanlinna took an active part role in musical events. It was while he was setting up a chamber music society in 1945 that he met the author and translator Sinikka Kallio-Visapaa. After his first marriage had been dissolved, he married Sinikka in 1958. They had a daughter Kristiina.

We have already mentioned the mathematical collaboration between Rolf Nevanlinna and his brother Frithiof. Walter Hayman writes of their collaboration in the 1920s [12]:-

Evenings and Sundays were excellent times for research, and Nevanlinna theory was greatly influenced by Rolf's discussions with Frithiof, which continued all their lives. They would walk up and down each side of a big square, talking mathematics.

They wrote papers together, for example: Über die Eigenschaften analytischer Funktionen in der Umgebung einer singulären Stelle oder Linie Ⓣ (1922), Über die Nullstellen der Riemannschen Zetafunktion Ⓣ (1924), and Über die Nullstellen der Riemannschen Zetafunktion Ⓣ (1925). Years later, they also collaborated in writing the book Absolute analysis (1959). An English translation was published in 1973. Nevanlinna coauthored the book Einführung in die Funktionentheorie Ⓣ (1965) with V Paatero. A J Lohwater begins a review:-

This book is a masterful account of the traditional first course in complex variables given at Helsinki University, and reflects in its first seven chapters the influence of E Lindelöf's introductory text.

As well as his textbooks and monographs, Nevanlinna published two books and over fifty articles which were intended to introduce mathematical ideas to non-mathematicians. One such example is his book Raum, Zeit und Relativität Ⓣ (1964). An example of his ideas on teaching mathematics, which became of considerable interest to him, is given in his paper Reform in teaching mathematics (1966).

Nevanlinna received many honours for his contributions. He was awarded the International Wihuri Prize for Scientists and Artists (1958) and the Henrik Steffens Prize for Nordic culture (1967). He was made an honorary member of the Deutsche Akademie (1938); the Finnish Mathematical Society (1955); the London Mathematical Society (1959); the Swiss Mathematical Society (1962); the Society of Actuaries of Finland (1965); the Teachers of Mathematics and Physics (1965); the Göttingen Academy of Sciences (1967); the Royal Swedish Academy of Sciences (1967); the Danish Academy of Sciences (1967); the German Academy of Sciences Leopoldina (1967); the Hungarian Academy of Sciences (1970); the Institut de France (1967); and the Finnish Academy of Science and Letters (1975). He was awarded honorary degrees by the universities of Heidelberg (1936), Bucharest (1942), Giessen (1952), Berlin (1955), Jyväskylä (1969), Glasgow (1969), Uppsala (1974) and Istanbul (1976). He was president of the International Mathematical Union from 1959 to 1962.

Nevanlinna continued to publish papers right up to the time of his death in 1980 from cancer. Since 1982, an award, the Rolf Nevanlinna Prize, is presented at the International Congress of Mathematicians. The one prize per congress is for young mathematicians dealing with the mathematical aspects of information science.