Bernhard Riemann (1826-1866) was the son of a poor country minister in northern Germany. He studied the works of Euler and Legendre while he was still in secondary school. It is said that he mastered Legendre's treatise on Theory of Numbers in less than a week! Riemann was shy and modest and was probably unaware of his own extraordinary abilities. In fact he went to the University of Gottingen (when he was nineteen) to study theology and (hence) become a minister himself. Fortunately before it was too late he realized his mistake and with the permission of his father switched to MATHEMATICS. The presence of the legendary C.F.Gauss made Gottingen the center of the mathematical world. But Gauss was remote and unapproachable, so after a while Riemann switched to University of Berlin. Once there he learned a great deal from Dirichlet and Jacobi. Two years later he returned to Gottingen, where he obtained his doctor's degree in 1851.
During the next eight years he endured debilating poverty and created his greatest works. In 1854 he was appointed as an "unpaid lecturer" which at the time was the necessary first step on the academic ladder. Gauss died in 1855, and Dirichlet was called to Gottingen as his successor. Dirichlet helped Riemann in every way he could and even arranged a small salary for him (around 1/10 of what a full professor would get). But Dirichlet also died in 1859 and Riemann was appointed as a full professor to replace him. Riemann's years of poverty were over, but his health was broken. In 1866, at the age of 39, he died of Tuberculosis in Italy where he used to go to escape the bitter cold weather of Germany.
Riemann had a short life and published comparatively little, but his works permanently altered the course of mathematics in analysis, geometry and number theory.
His first published paper was his celebrated dissertation of 1851 on the general theory of functions of a complex variable. Riemann's fundamental aim here was to free the concept of an analytic function from any dependence on explicit expressions such as power series, and to concentrate instead on general principles and geometric ideas. In doing so he used/created concepts such as Cauchy-Riemann equations, Riemann surfaces etc. Gauss was rarely enthusiastic about achievements of his contemporaries, but in his official report to the faculty he warmly praised Riemann's work: "The dissertation submitted by Herr Riemann offers convincing evidence of the author's thorough and penetrating investigations in those parts of the subject treated in the dissertation, of a creative, active, truly mathematical mind, and of a gloriously fertile originality."
Riemann later applied these ideas to the study of hyper geometric and Abelian functions. The geometric reasoning and physical insight employed by Riemann in this work of his can be considered as truly the beginning of Topology (a rich field of geometry concerned with those properties of figures that remain unchanged by continuous deformations).
In 1854, on the way to becoming the "Unpaid lecturer" Riemann was required to submit a probationary essay. And his response was another pregnant work whole influence is indelibly stamped on mathematics of our time. This was his work on the concept of the so called Riemann Integral which now appears on most textbooks on calculus. In the same paper he generalized the Dirichlet Criteria for the validity of the Fourier expansions. Cantor's famous theory of sets was directly inspired by this work. The same ideas also led to the concept of Lebesgue Integral and other even more general types of Integration. Riemann's pioneering work thus led to another new branch of mathematics, the theory of functions of a real variable.
The Riemann rearrangement theorem in the theory of infinite series was an incidental result in the paper just described. As an example: 1-1/2+1/3-1/4+... = log 2 and 1+1/3-1/2+1/5+1/7-1/4+... = 3/2 log 2
It is apparent that these two series have different sums but the same terms. Riemann proved that it is possible to rearrange the terms of 'any' conditionally convergent series in such a manner that the new series will converge to an arbitrary preassigned sum or diverge to infinity or minus infinity!
In addition to his probationary essay, Riemann was also required to present a trial lecture to the faculty before he could be appointed to his "unpaid lecturership". It was the custom for the candidate to offer three titles, and the head of his department usually accepted the first. However, Riemann rashly listed as his third topic the foundations of geometry, a profound subject on which he was unprepared but which Gauss had been turning over in his mind for 60 years. Naturally, Gauss was curious to see how this particular candidate's "gloriously fertile originality" would cope with such a challenge, and to Riemann's dismay he designated this as the subject of the lecture. Riemann quickly tore himself away from his other interests at the time - "my investigations of the connection between electricity, magnetism, light and gravitation" - and wrote his lecture in the next two months. The result was one of the great classical masterpieces of mathematics, and probably the most important scientific lecture ever given. It is recorded that even Gauss was surprised and enthusiastic.
Riemann's lecture presented in nontechnical language a vast generalization of all known geometries, both Euclidean and non-Euclidean. This field is now called Riemannian Geometry; and apart from its great importance in pure mathematics, it turned out 60 years later to be exactly the framework for Einstein's general theory of relativity. Like most of the great ideas of science, Riemannian geometry is quite easy to understand if we set aside the technical details and concentrate on its essential features. Gauss had earlier discovered the intrinsic differential geometry of curved surfaces. If a surface embedded in three dimensional space is defined parametrically by three functions x=x(u,v), y=y(u,v), and z=z(u,v), then u and v can be interpreted as the coordinates of the points on the surface. The distance ds between any two nearby points (u,v) and (u+du,v+dv) is given by Gauss's quadratic differential form:
ds = Edu^2 + 2Fdudv + Gdv^2,
where E, F and G are certain functions of u and v. This differential form makes it possible to find lengths of curves on the surface, shortest path between two points on the surface (geodesic), and the Gaussian curvature of the surface at any point - all in total disregard of the surrounding space. Riemann generalized this by discarding the idea of a surrounding Euclidean space and introducing the idea of a continuous n-dimensional manifold of points (x(1), x(2),...,x(n)). He then imposed an arbitrarily given distance (or metric) ds between nearby points (x(1), x(2),...,x(n)) and (x(1)+dx(1), x(2)+dx(2),...,x(n)+dx(n)) by means of a quadratic differential form:
ds = g(i,j)x(i)x(j) ---> summed over i and j
where g(i,j)'s are suitable functions of x(1), x(2),...x(n) and different systems of g(i,j)'s define different Riemannian geometries on the manifold under discussion. His next steps were to examine the idea of curvature for these Riemannian manifolds and to investigate the special case of constant curvature. All of this depends on massive computational machinery, which Riemann mercifully omitted from his lecture but included in a posthumous paper on heat conduction. In that paper he explicitly introduced the Riemann curvature tensor, which reduces to the Gaussian curvature when n=2 and whose vanishing he showed to be necessary and sufficient for given quadratic metric to be equivalent to a Euclidean metric. From this point of view, the curvature tensor measures the deviation of the Riemannian geometry from Euclidean geometry. Einstein has summarized these ideas in a single statement: "Riemann's geometry of an n-dimensional space bears the same relation to Euclidean geometry of an n-dimensional space as the general geometry of curved surfaces bears to the geometry of the plane."
The physical significance of geodesics appears in its simplest form as the following consequence of Hamilton's principle in the calculus of variations: if a particle is constrained to move on a curved surface, and if no force acts on it, then it glides along a geodesic. A direct extension of this idea is the heart of Einstein's general theory of relativity, which is essentially a theory of gravitation. Einstein conceived the geometry of the space as a Riemannian geometry in which the curvature and geodesics are determined by the distribution of matter; in this curved space, planets move in their orbits around the sun by simply coasting along geodesics instead of being pulled into curved paths by a mysterious force of gravity whose nature no one has ever really understood.
In 1859 Riemann published his only work on the theory of numbers, a brief but exceedingly profound paper of less than 10 pages devoted to the prime number theorem. This mighty effort started tidal waves in several branches of pure mathematics, and its influence will probably still be felt a thousand years from now. His starting point was a remarkable identity discovered by Euler over a century earlier: if s is a real number greater than 1, then: 1 + 1/2^s + 1/3^s +... = 1/[(1-1/2^s)(1-1/3^s)(1-1/5^s)(1-1/7^s)...] where the expansion on the right denotes the product of the numbers 1/(1-1/p^s) for all primes p. To understand how this identity arises, we note that 1/(1-x) = 1 + x + x^2 + x^3 + ... for |x| < 1, so for each p we have: 1/(1-1/p^s) = 1 + 1/p^s + 1/p^2s + 1/p^3s + ... On multiplying these series for all primes p and recalling that each integer >1 is uniquely expressible as a product of powers of different primes, we get the left hand side of the identity. The sum of the series on the left is evidently a function of the real variable s>1, and the identity establishes a connection between the behaviour of this function and properties of the primes. Euler himself exploited this connection in several ways, but Riemann perceived that access to the deeper features of the distribution of primes can only be gained by allowing s to be a complex variable. He denoted the resulting function as C(s) [read zeta(s)], and it has since been known as the Riemann zeta function:
C(s) = 1 + 1/2^s + 1/3^s + ..., s = x + iy.
In his paper he proved several important properties of this function, and in a sovereign way simply stated a number of others without proof. During the century since his death, many of the finest mathematicians in the world have exerted their strongest efforts and created rich new branches of analysis in attempts to prove these statements. The first success was achieved in 1893 by J. Hadamard, and with one exception every statement has since been settled in the sense Riemann expected. The exception is the famous Riemann hypothesis: that all zeros of C(s) in the strip 0 < x < 1 lie on the central line x = 1/2. It stands today as the most important unsolved problem of mathematics, and is probably the most difficult problem that the mind of man has ever conceived. In a fragmentary note found among his posthumous papers, Riemann wrote that these theorems "follow from an expression for the function C(s) which I have not yet simplified enough to publish." Writing about this fragment in 1944, Hadamard remarked with justified exasperation, "We still have not the slightest idea of what the expression could be." He adds the further comment:"In general, Riemann's intuition is highly geometrical; but this is not the case for his memoir on prime numbers, the one in which that intuition is the most powerful and mysterious."