Lecture 36. Riemann, one of the MostImportant Mathematicians

Figure 36.1 Bernhard Riemann and the Loneburg.

Riemanns name connects many mathematical concepts: Riemann sphere,Cauchy-Riemann equation, Riemann hypothesis, Riemann integral, Riemanniangeometry, Riemann surface, Riemann mapping theorem, etc. Also, integrationwas first rigorously formalized, using limits, by Riemann.

Bernhard Riemann Bernhard Riemann (1826-1866) was born in the village of Breselenznear Hanover in Germany.

His father, a Lutheran minister, acted as teacher to educate his six children. In 1840Bernhard entered directly into the third class at the gymnasium (a sort of prep school) atHanover. Bernhard Riemann was a good, hard working pupil, but not outstanding, at theclassical subjects such as Hebrew and theology.

At the age of 16, Riemann transferred to the gymnasium at Luneburg, where his remark-able mathematical talent was noticed. The director of the Gymnasium, Schmalfuss, allowedBernhard to study mathematics texts from his own library. On one occasion, he lent him

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Legendres book on the theory of numbers. Six days later Riemann returned the 859-pagebook, saying, That was a wonderful book! I have mastered it. And he had.

In 1846, Riemann entered in the Gottingen University to study philology and theology.In accordance with his fathers wishes, he began in the faculty of theology. His mathematicalstudies at first were probably just a relaxation, but it was destined to be the chief business ofhis life; he soon transferred to pursue science and mathematics. He attended many lectures,including one on the method of least squares by Carl Friedrich Gauss.

In 1847, Riemann went to Berlin, attracted there by brilliant mathematicians such as P.G. L. Dirichlet, Karl Gustav Jacobi, J. Steiner and F. G. M. Eisenstein. It was during thisperiod when he attended Dirichlets and Jacobis lectures, that Riemann gradually formedhis ideas on the theory of functions of a complex variable which led to most of his greatdiscoveries.

Figure 36.2 Riemann and the Gottingen University.

In 1849 he returned to Gotingen to complete this training for a doctorate. His these of1851, written under Gauss, dealt with surfaces which are called Riemann surfaces today. Hismemoir excited the admiration of Carl Friedrich Gauss, and at once establish his reputationas a mathematician of first rank.

In 1853, Gauss asked Riemann to prepare a Habilitationsschrift which was the highestacademic qualification a scholar can achieve by his own in Germany. Of the three themeswhich he suggested for his trial lecture, On the Hypotheses Which Form the Foundationof Geometry was chosen by Gauss, who was clearly very interested and curious to hearwhat so young a man had to say on this difficult subject. Riemanns inaugural lecture was

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extraordinary successful which opened a new world of geometry. It is said that nobodypresent understood Riemanns approach to geometry except Gauss and that even Gaussmight be perplexed.

In 1855 Gauss died. His successor Dirichlet, who along with others, made an effort toobtain Riemanns nomination as extraordinary professor, but was not successful. Instead, agovernment stipend of 200 thalers was given to Riemann; though not much, it was of greatimportance in his circumstances. But shortly, Riemman was hit by the deaths of his fatherand his eldest sister. He continued to extend his results in his doctoral dissertation.

Riemanns health had never been strong since he was a child. It got worse under thestrain of work and he broke down altogether, and had to stay with his friends Ritter and R.Dedekind for a while.

In 1857, Riemann returned to Gottingen where he was made extraordinary professor.As a result, his salary raised to 300 thalers. Sadly in quick succession he lost his brotherWilhelm and another sister.

In 1859 he lost his friend Dirichlet; but his reputation was now so well established thathe was at once appointed to succeed him. In 1860 he visited Paris, and was met with awarm reception there.

He married Elise Koch in June 1862, but the following month he had an attack of pleurisy.

After staying in Italy for the benefit of his health, Riemann returned to Gottingen inJune 1863. But he caught cold, and had to go back to Italy in August. In November1865 he returned again to Gottingen. He got weaker and weaker. In June 1866 Riemannreturned once more to Italy. Here his strength rapidly ebbed away, but his mental facultiesremained brilliant to the last. On the 19th of July 1866 he was working at his last unfinishedinvestigation on the mechanism of the ear. The day following he died of tuberculosis at theearly age of 39.

New geometry of Riemann Based upon the discovery of non-Euclidean geometrywhich was modelled by Neltrami, Klein and Poincare, Riemann was thinking about a newtype of non-Euclidean geometry. Riemann believed that observation on physical space hadnot confirmed the existence of parallel lines, and that one could assume: every pair of lineswould meet at some point. Riemanns good illustration is the surface of a sphere, with greatcircles regarded as lines.

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In his famous inaugural lecture on the foundation of geometry, Riemann discussed inparticular the ideas now called manifolds, Riemannian metric, and curvature. He constructedan infinite family of non-Euclidean geometries by giving a formula for a family of Riemannianmetrics on the unit ball in Euclidean space. He proposed that a space M with a metric gdefines a geometry. Today such metric g is called Riemannian metric, such pair (M, g) iscalled a Riemannian manifold with respect to the metric, and geometry under this point ofview is called Riemannian geometry.

As a result, the classical Euclidean geometry is a special case of Riemannian geometrywith curvature 0; the standard non-Euclidean geometry is a special case of the Riemanniangeometry with curvature 1, and the geometry of sphere is a special Riemannian geometrywith curvature +1. And there are many more different kinds of geometry. Later develop-ment has shown that Riemanns work was far-reaching, with his theorems holding for allgeometries.

Figure 36.3 Riemann and Riemann surfaces.

In his short life Riemann published only a relatively small number of papers but each ofthem was important and several have opened entirely new and productive fields. Riemannswork made huge contributions and influences in mathematics.

Riemannian integral and real analysis Riemann put the definite integral on afirmer basis with a generalization now called the Riemann integral.

Riemann invented a function which was discontinuous at an infinite number of pointsin an interval, but the integral of which still existed.

Riemann gave an example of a continuous function without derivatives (Weierstrassgave such an example which was published in 1875). Mathematicians regarded such

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functions as pathological ones and refused to take them seriously; modern analysishas shown how natural such functions are and how deep Riemanns insight was.

Riemannian geometry In his famous Habilitationsschrift in 1854,On the hypothe-ses which underlie geometry, Riemann developed his theory of higher dimensions. Itis one of the most important works in geometry.

Riemanns work opened a new field called Riemannian geometry. Riemann foundthe correct way to extend the theory of differential geometry of surfaces into higherdimensional cases.

Riemann suggested a new form of non-Euclidean geometry that later became thegeometry of Einsteins general theory of relativity.

Complex geometric analysis Riemanns works opened up research areas combin-ing analysis with geometry, which have become the fields of Riemannian geometry,algebraic geometry, and complex manifold theory.

Number theory In a single short paper, which was the only one he published onthe subject of number theory, Riemann introduced the Riemann zeta function andestablished its importance for understanding the distribution of prime numbers. Hemade a series of conjectures about properties of the zeta function, one of which is thewell-known Riemann hypothesis.

Algebra Riemann developed the theory of quadratic forms.

Applied Dirichlet principle Riemann applied the Dirichlet principle to attackmany important problems; this was later seen to be a powerful heuristic rather thana rigorous method. Its justification took at least a generation.

Figure 36.4 Riemann

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Riemann hypothesis Riemanns paper, Ueber die Anzahl der Primzahlen unter einergegebenen Grosse (On the number of primes less than a given quantity), was first published in1859. In this six-page paper, Riemann introduced radically new ideas to the study of primenumbers, ideas which led to independent proofs by Hadamard and de la Vallee Poussin ofthe prime number theorem in 1896. This theorem, first conjectured by Gauss when he wasa young man, states that the number of primes less than x is asymptotic to x

log(x).

Figure 36.5 The manuscript in which he Riemann formulated the famous Riemann hypothesis.

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Riemann gave a formula for the number of primes less than x in terms the integral of1

log(x)and the roots (zeros) of the zeta function, defined by

(s) = 1 +1

2s+

1

3s+

1

4s+ ....

He also formulated a conjecture about the location of these zeros, which fall into two classes:the obvious zeros -2, -4, -6, etc., and those whose whose real part lies between 0 and 1.Riemanns conjecture was that the real part of the non-obvious zeros is exactly 1/2. Thatis, they all lie on a specific vertical line in the complex plane.

Riemann checked the first few zeros of the zeta function by hand. They satisfy hishypothesis. By now over 1.5 billion zeros have been checked by computer. Very strongexperimental evidence indicates that it could be true, but there is no mathematical proof.

The Riemann Hypothesis is probably the hardest unsolved problem in all of mathematics,and one of the most important. It was one of the famous Hilbert problems, the number eightof twenty-three. It is also one of the seven Clay Millennium Prize Problems.

The great mathemetician, David Hilbert, was once asked: if you could go to sleep for500 years, what question would you first ask when you woke up? His answer: Has anyoneproved the Riemann Hypothesis?

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