Georg Friedrich Bernhard Riemann Ted Sheehy 1 Personal Life Georg Friedrich Bernhard Riemann is one of the most influential mathematicians the field has seen. In his short 40 year life he published works that changed the way mathematicians view problems and have provided a basis for modern mathematics. He posed problems that influenced great thinkers for centuries. One of which a century and a half later still has not been solved. Riemann was born in 1826 in Breselenz Germany. His family was not one of mathematical geniuses. His father was a Lutheran minister, as was his grandfa- ther and many others in his family. He had five siblings none of them displayed the same genius that quickly became apparent in Riemann. He excelled in school and became so far ahead of his peers he had to be given a personal tutor. This tutor was intended to teach Riemann subjects the rest of the children at his age could not understand. However, after a short time the tutor noted that he was learning from Riemann, not the other way around. At 14 Riemann left home and moved in with his grandmother to attend a well respected school in Hanover. This was hard on the young, shy student. He began to struggle with loneliness that would plague him for the rest of his short life [1]. Later Riemann enrolled in the University of Gottingen. He planned to follow the lineage of ministers that preceded him. He studied theology and philosophy. That is until he started attending the lectures of well known mathematician Carl Friedrich Gauss [1]. Gauss is a well respected and very influential mathematician. Some consider him the greatest mathematician of all time. Many later mathematicians, includ- ing Riemann, used his work as a basis for their own discoveries. At 21 Gauss wrote Disquisitiones Arithmeticae on number theory. This work is considered to be the foundation for modern mathematics. It helped establish Mathematics as its own field of study, separate from physics and astronomy, which at the time, was a major field of study. Anyone who could track and understand the movements of celestial bodies was highly respected. Astronomers at the time were tracking an asteroid call Ceres. They lost track of it and were unable to rediscover it. Gauss was able to use his newly developed methods to predict the location of the asteroid and rediscover it. This accomplishment gained tremen- dous respect for both Gauss as an individual and for the field of mathematics as a whole [2]. 1 By the time Riemann began to study at Gottingen Gauss was in his seventies. The old and ailing Gauss encouraged Riemann to transfer to the University of Berlin where he could study with a new generation of talented mathematicians. Riemann followed the advice and transferred to the University of Berlin where he graduated in 1849. He later returned to Gottingen to study under Gauss and complete his doctoral thesis, Foundations of a General Theory of Functions of a Complex Variable [1]. 2 Riemann Sums At the time mathematics was a fairly small field and there were very few job opportunities. Riemann struggled to find work so he wrote another dissertation called a Habilitation to help him secure a teaching job at a University. Rie- mann's Habilitation was titled On the representability of a function by means of a trigonometric series. At the time is was already known and understood that a definite integral was the sum of infinitely many smaller sums. That is, the area under a curve can be calculated by adding the area of infinitely many small shapes under the curve. The figure below represents this basic idea. [3] This understanding of integrals is fundamental to the study of calculus and plays a huge role in mathematics. However at the time there was a major hole in the understanding. Mathematicians in Riemann's age understood the principle displayed in the image but they preferred to use anti-derivatives and the Funda- mental Theorem of Calculus when dealing with integrals. The problem was, this 2 method only works on functions that have elementary anti-derivatives. Math- ematicians sought a method that would apply to a wider range of functions. Riemann would find that solution [1]. Riemann developed a method of calculating the infinite sums using any sub- intervals. The basic formula is: P1 b−a limx!1 n=1 f(x)∆x where ∆x = n This method allows us to calculate integrals for a larger array of functions. It has become known as a Riemann sum and is a fundamental part of any calculus class. 3 Riemann Geometry Once Riemann completed his second dissertation he still was unable to get a teaching job. He had to also complete a Habilitation lecture. Gauss chose On the hypotheses that underlie the foundations of geometry for Riemann to present on. This is a field that Gauss himself had been working in. He was trying to survey triangles who's angles did not add up to 180 degrees. This lecture would eventually give birth to what many now call Riemann Geometry. The concepts discussed in the lecture are fairly complex. But we can get a simplified understanding if we examine Riemann's idea of curvature and see how it challenged the conventions of Euclidean geometry. Riemann re- examined basic ideas like, \the sum of the angles of a triangle is 180" and \the shortest distance between two points is a straight line". Riemann applied the concepts of curvature and multidimensional spaces. If we apply curvature these basic rules of Euclidean Geometry can be broken. If you take a piece of paper and draw two points the shortest distance be- tween them is obviously a straight line. But what if we do the same thing on a cylinder or a sphere? Now the shortest distance along the surface is not a straight line. It is actually a curved line that follows the curvature of the surface. This idea can also be applied to triangles. The image below represents the idea of positive and negative curvature. We can see that in a space with positive curvature, each angle of the triangle is larger than 60 degrees and the sum of the angles of the triangle is greater than 180. In a space with negative curvature the angles are each less than 60 degrees and the sum is less than 180. 3 [4] Riemann's ideas would become the basis which Einstein used to build his famous theory of relativity [1, 2]. 4 Mental Health Once Riemann finished his Habilitation lecture he was able to start teaching. However he was unable to land a permanent chair as a professor. Instead he survived by giving lectures to small groups of students. This was not a high paying occupation and he struggled to support himself financially. To make matters worse, Riemann's mentor Gauss and Riemann's father both died in the same year. This combined with the stress of his financial situation caused Riemann to have a mental breakdown. Riemann left the world of teaching and mathematics. He fled to the moun- tains where spent time with nature and worked to ease his troubled disposition. Eventually, after two years of recovering in nature, Riemann was ready to return to mathematics. In 1857 he began teaching as an assistant professor. Finally Riemann had a steady job and his finances in order. Just as things started to look up for him, his brother died and he was forced to care for his three unmarried sisters. This took a huge toll on Riemann. He began to show signs of tuberculous, a nagging illness that would eventually take his life [1]. 5 Riemann Hypothesis In 1859 Riemann became the chair of mathematics at the University of Gottin- gen and was elected to the Berlin Academy of Science. This success inspired him to write another paper. This one was entitled On the Number of Prime Numbers Less Than a Given Magnitude. In this paper Riemann would would outline what is now called the Riemann Hypothesis. This is one of the most fa- mous unsolved problems of mathematics. Mathematicians have been struggling to prove (or disprove) this hypothesis for over a century and a half to no avail. There is currently a $1,000,000 prize for anyone that can prove the hypothesis. 4 The very basic idea of Riemann's Hypothesis says that the only non-trivial solutions to the Zeta Function, shown below, lie along the line where s is a complex number with its real part equal to 1=2 [1]. 1 X 1 1 1 1 ζ(s) = = + + ::: ns 1s 2s 3s n=1 6 Decline and Death In 1862 Riemann married Elise Koch, but almost immediately his health began to decline. He fled to Italy during the winter, hoping the warm weather would help his condition. After a few years of struggling with illness Riemann eventually died on July 20, 1866, just before his 40th birthday. He left behind his wife Elise and their daughter Ida [1, 2]. References [1] Hawking, Stephen. God Created the Integers: the Mathematical Break- throughs That Changed History. Running Press, 2007. [2] Riemann, Bernhard, and Jurgen Jost. On the Hypotheses Which Lie at the Bases of Geometry. Birkhauser, 2016. [3] KSmrq. \Intergral Approximations." Wikimedia Commons, 7 July 2007, commons.wikimedia.org/wiki/File:Integralapproximations.svg. [4] \Geometry of the Universe." University of Oregon, abyss.uoregon.edu/ js/21stcenturyscience/lectures/lec21.html. 5.