Bernhard Riemann

How to use this course

Bernhard Riemann published only a small number of papers in a working life of about fifteen years. Almost every one of them opened a field. Complex analysis, differential geometry, the theory of algebraic curves, analytic number theory and the theory of hyperbolic partial differential equations all carry his name in their basic vocabulary. This course follows those papers in order and sets them against the story of a modest, often ill man who was supported at every step by Carl Friedrich Gauss, Peter Gustav Lejeune Dirichlet and Richard Dedekind.

Each technical lesson opens with a paragraph for the general reader, then gives precise statements, proof sketches and exercises with folded solutions. Riemann often stated more than he proved. Where later mathematicians supplied the proofs, the lesson says who and when.

Timeline

Breselenz and Lüneburg, 1826–1846

Georg Friedrich Bernhard Riemann was born on 17 September 1826 in Breselenz, a village near Dannenberg in the Kingdom of Hanover. His father, Friedrich Bernhard Riemann, was a Lutheran pastor. His mother, Charlotte Ebell, died in 1846. Bernhard was the second of six children. The family was poor, and most of his siblings died young.

His father taught him at home until he was about ten, with help from a local teacher. In 1840 he went to the Lyceum in Hanover and lived with his grandmother. When she died in 1842 he moved to the Johanneum, the grammar school in Lüneburg. He was intensely shy and homesick, and he wrote often to his family. The director of the gymnasium noticed his gift and lent him books from his own library. He lent Riemann Adrien-Marie Legendre’s treatise on number theory, about 900 pages long, and Riemann is said to have read it in six days. The episode marks the first contact with a subject he would transform.

The family expected him to follow his father into the ministry, and he shared the expectation. His faith remained serious all his life. But by the time he left school his real interest was clear.

Göttingen and Berlin, 1846–1851

In spring 1846 Riemann enrolled at the University of Göttingen as a student of theology and philology. He attended mathematics lectures by Moritz Stern and by Gauss, who then taught only elementary courses. Within a year he asked his father’s permission to switch to mathematics, and received it.

Göttingen offered little advanced teaching, so in 1847 he moved to Berlin. There he heard Carl Gustav Jacobi on mechanics and algebra, Jakob Steiner on geometry, Gotthold Eisenstein on elliptic functions, and Dirichlet on number theory, definite integrals and partial differential equations. Dirichlet became the decisive influence: his blend of intuition and rigour, and his habit of thinking with as little calculation as possible, became Riemann’s own. Eisenstein, three years older, discussed complex function theory with him; their approaches differed sharply, with Eisenstein working through formulas and Riemann through general principles.

In 1849 he returned to Göttingen. He took part in the physics seminar of Wilhelm Weber and assisted him for about eighteen months. Physics stayed central to his thinking. He hoped all his life to find a unified mathematical description of light, electricity, magnetism and gravitation.

The doctorate

His thesis, Grundlagen für eine allgemeine Theorie der Functionen einer veränderlichen complexen Grösse, was examined on 16 December 1851 with Gauss as advisor. It defined a complex function by its local behaviour, through what we now call the Cauchy–Riemann equations, introduced multi-sheeted surfaces spread over the plane, and stated the mapping theorem of Lesson 04. Gauss’s report was unusually warm and credited the author with a gloriously fertile originality.

The habilitation, 1851–1854

To teach at a German university Riemann needed the habilitation: a second thesis and a trial lecture. He spent about two and a half years on the thesis, Über die Darstellbarkeit einer Function durch eine trigonometrische Reihe. It set out to answer Dirichlet’s question of which functions can be represented by Fourier series, and along the way it gave the modern definition of the integral (Lesson 05).

For the lecture, the candidate proposed three topics and the faculty chose one. Riemann offered two subjects in mathematical physics and, third, the foundations of geometry, which he had prepared least. Gauss chose geometry. Riemann was exhausted from his physics work with Weber and spent weeks in a state close to breakdown before he could write. On 10 June 1854 he delivered Über die Hypothesen, welche der Geometrie zu Grunde liegen to the faculty, using almost no formulas. Gauss, then 77, spoke to Weber afterwards about the depth of its ideas with rare enthusiasm.

Riemann became a Privatdozent, paid only by student fees. His first course in the winter of 1854–55 on partial differential equations attracted eight students. Neither the habilitation thesis nor the lecture was published in his lifetime. Dedekind brought both into print in 1867 and 1868.

Complex functions and Riemann surfaces

For the general reader. Before Riemann, a function was usually a formula. Riemann defined a complex function by a local rule, differentiability in the complex sense, and then let the function tell him where it lived. A function such as the square root takes two values, so he spread it over a surface made of two sheets of the plane glued together. The shape of that surface, its number of holes, controls the function.

Local definition

Write \(w=f(z)=u+iv\) with \(z=x+iy\). Riemann required that the derivative \(dw/dz\) be independent of the direction of \(dz\). This is equivalent, for differentiable \(u,v\), to

Then \(\Delta u=\Delta v=0\): real and imaginary parts are harmonic. Geometrically, \(f\) preserves angles where \(f'\neq0\). Riemann’s thesis emphasizes this conformal picture, which Gauss had studied for maps of surfaces.

The Dirichlet principle

To construct functions, Riemann needed harmonic functions with prescribed boundary values. He argued that among all functions \(u\) on a region \(\Omega\) with given values on \(\partial\Omega\), one minimizes the Dirichlet integral \(D[u]=\iint_\Omega |\nabla u|^2\,dx\,dy\), and that the minimizer is harmonic. He named the argument after Dirichlet, from whose lectures he had learned it. In 1870 Karl Weierstrass showed that a positive functional need not attain its infimum, so the existence of a minimizer is not automatic. Hilbert rescued the principle around 1900 under suitable hypotheses, and the modern theory of Sobolev spaces makes it rigorous.

Riemann’s proof used the Dirichlet principle and assumed a smooth boundary. William Fogg Osgood gave the first complete proof in 1900. The standard modern proof, due to Leopold Fejér and Frigyes Riesz, maximizes \(|f'(z_0)|\) over a normal family of injective maps into the disc.

Surfaces

The function \(w=\sqrt z\) has two values at each \(z\neq0\). Take two copies of the plane slit along the negative real axis. Glue the upper edge of each slit to the lower edge of the other. The result is a connected surface on which \(\sqrt z\) is single-valued, with a branch point over \(0\). More generally, an algebraic equation \(F(z,w)=0\) defines a surface spread over the \(z\)-sphere, and Riemann classified such surfaces by their connectivity, the number now called the genus. Hermann Weyl’s book Die Idee der Riemannschen Fläche (1913) gave the intrinsic definition used today: a connected one-dimensional complex manifold.

The integral and trigonometric series

For the general reader. Fourier claimed that every function is a sum of sines and cosines. Dirichlet proved it for well-behaved functions and asked how far the claim extends. To even ask the question for wild functions, Riemann first had to say what the integral of a wild function means. His definition, by sums over finer and finer partitions, is the one taught in every calculus course.

The integral

For a bounded \(f\) on \([a,b]\) and a partition \(a=x_0 \lt x_1 \lt \cdots \lt x_n=b\) with sample points \(\xi_k\in[x_{k-1},x_k]\), form the sum \(S=\sum_k f(\xi_k)(x_k-x_{k-1})\). If \(S\) tends to a limit as the mesh \(\max(x_k-x_{k-1})\) tends to 0, independently of the choices, that limit is \(\int_a^b f\).

Lebesgue later recast this as: \(f\) is Riemann integrable iff its set of discontinuities has measure zero. Riemann illustrated the strength of his criterion with

where \(((x))\) is \(x\) minus the nearest integer, and 0 at half-integers. The series converges uniformly. Its sum is discontinuous at every rational number with even denominator in lowest terms, a dense set, yet only finitely many jumps exceed any given size, so \(f\) is integrable.

Trigonometric series

Proof sketch. For the indicator of an interval \([c,d]\), \(\int_c^d\sin nx\,dx=(\cos nc-\cos nd)/n\to0\). Step functions are finite combinations of these. By the criterion, an integrable \(f\) is within \(\varepsilon\) in \(L^1\) of a step function \(s\), and \(|\int(f-s)\sin nx|\le\varepsilon\). Riemann used the lemma to prove the localization principle: whether the Fourier series of \(f\) converges at \(x\) depends only on the values of \(f\) near \(x\). He also studied series \(\sum(a_n\cos nx+b_n\sin nx)\) with \(a_n,b_n\to0\) that are not Fourier series, by integrating twice term by term. Georg Cantor’s work on uniqueness of such series, which led him to set theory, starts from Riemann’s methods.

Proof. The positive terms and the negative terms each form a divergent series, and \(a_n\to0\). Add positive terms until the partial sum first exceeds \(L\), then negative terms until it first drops below \(L\), and repeat. Each overshoot is at most the size of the last term used, which tends to 0.

The hypotheses of geometry

For the general reader. Euclid’s geometry was thought to be the necessary structure of space. Riemann proposed that geometry is a set of hypotheses to be tested by physics. A space of any number of dimensions can carry a rule for measuring small distances, and that rule can vary from place to place. From it one can compute curvature. Sixty years later Albert Einstein used exactly this framework for gravity.

Manifolds and metrics

Riemann began with an \(n\)-fold extended magnitude, a space in which a point is fixed by \(n\) continuous coordinates. He assumed that the length of a curve is computed from an infinitesimal line element and that this line element is the square root of a positive definite quadratic form:

He noted that other choices, such as the fourth root of a quartic form, are possible. Today metrics of that kind are called Finsler metrics.

Counting

The symmetric matrix \((g_{ij})\) has \(n(n+1)/2\) component functions. A change of coordinates has \(n\) free functions. Riemann concluded that \(n(n+1)/2-n=n(n-1)/2\) functions describe the geometry independently of coordinates, and he identified them with the curvatures of the two-dimensional surfaces through each point, one for each pair of directions. This is the idea of sectional curvature.

Riemann wrote down the curvature expression, in different notation, in his 1861 Paris prize memoir on heat conduction, where he asked when a quadratic form can be transformed into one with constant coefficients. The lecture itself gives the metric of a space of constant curvature \(\alpha\):

For \(\alpha \gt 0\) this is the round sphere in stereographic coordinates; for \(\alpha \lt 0\) it is hyperbolic space in the ball model. Riemann also distinguished between space being unbounded and being infinite: a sphere of positive curvature is unbounded but finite.

After Riemann

Elwin Christoffel and Rudolf Lipschitz developed the analytic theory in 1869–70. Gregorio Ricci-Curbastro and Tullio Levi-Civita built the tensor calculus in 1900. Einstein, with Marcel Grossmann’s help, made the metric of four-dimensional spacetime the gravitational field in 1915.

Abelian functions and Riemann–Roch

For the general reader. Integrals such as \(\int dx/\sqrt{1-x^4}\) had occupied Euler, Abel and Jacobi. Riemann saw that these integrals are best understood on the surface of the underlying algebraic equation, and that one number, the genus, controls everything: how many independent integrals exist, how many functions with given poles exist, and what the surface looks like.

Genus and differentials

A compact Riemann surface \(X\) is topologically a sphere with \(g\) handles; \(g\) is its genus. Riemann showed that \(X\) has exactly \(g\) linearly independent holomorphic differentials \(\omega_1,\dots,\omega_g\) (abelian integrals of the first kind). If \(X\to\mathbb P^1\) is a branched covering of degree \(d\) with ramification indices \(e_p\), then

Riemann had this count for plane curves; Adolf Hurwitz proved the general formula for maps between surfaces in 1891.

Functions with prescribed poles

For a divisor \(D=\sum n_pp\) let \(\ell(D)\) be the dimension of the space of meromorphic functions \(f\) with \(\operatorname{div}f+D\ge0\), and let \(K\) be the divisor of a meromorphic differential. Riemann proved in 1857 that \(\ell(D)\ge\deg D-g+1\). His student Gustav Roch identified the correction term in 1865.

Riemann’s argument. A function with at most simple poles at \(m\) points \(p_1,\dots,p_m\) is, up to a constant, a combination of abelian integrals of the second kind with poles at the \(p_j\). That gives \(m+1\) parameters. The function must be single-valued, so its \(2g\) periods must vanish; the residue-free construction already kills \(g\) of them, leaving \(g\) linear conditions. Hence \(\ell\ge m+1-g\). The conditions are dependent exactly when some holomorphic differential vanishes at all the \(p_j\), which is Roch’s term.

Theta functions

Choose a symplectic basis \(a_1,\dots,a_g,b_1,\dots,b_g\) of loops and normalize \(\int_{a_j}\omega_k=\delta_{jk}\). The period matrix \(\Omega_{jk}=\int_{b_j}\omega_k\) satisfies Riemann’s bilinear relations: \(\Omega\) is symmetric and \(\operatorname{Im}\Omega\) is positive definite. Riemann used these to define

which converges because of the positivity. He solved the Jacobi inversion problem by showing that the zeros of \(\theta\) composed with the Abel map recover \(g\) points on \(X\), and found the vanishing locus of theta (Riemann’s vanishing theorem). He also counted \(3g-3\) complex parameters, or moduli, for surfaces of genus \(g\ge2\).

The zeta function and the primes

For the general reader. In November 1859 Riemann sent the Berlin Academy an eight-page paper, his only one on number theory, to thank it for electing him. He showed that the irregular distribution of the primes is governed by the zeros of a single function of a complex variable. He conjectured that all the interesting zeros lie on one line. That conjecture, the Riemann hypothesis, is still open.

Euler product and continuation

For \(\operatorname{Re}s \gt 1\), \(\zeta(s)=\sum_{n\ge1}n^{-s}=\prod_p(1-p^{-s})^{-1}\), the product over primes being Euler’s. Riemann treated \(s\) as a complex variable. Starting from \(\Gamma(s/2)\pi^{-s/2}n^{-s}=\int_0^\infty e^{-\pi n^2x}x^{s/2-1}\,dx\) and summing over \(n\), he expressed \(\zeta\) through the theta function \(\vartheta(x)=\sum_{n\in\mathbb Z}e^{-\pi n^2x}\), which satisfies \(\vartheta(1/x)=\sqrt x\,\vartheta(x)\).

The zeros of \(\zeta\) are the trivial zeros at \(-2,-4,-6,\dots\), where \(\Gamma(s/2)\) has poles, and the non-trivial zeros \(\rho\), which are the zeros of \(\xi\) and lie in the strip \(0\le\operatorname{Re}s\le1\). They are symmetric about the real axis and about the line \(\operatorname{Re}s=\tfrac12\). Riemann stated that the number with \(0 \lt \operatorname{Im}\rho \lt T\) is about \(\frac{T}{2\pi}\log\frac{T}{2\pi}-\frac{T}{2\pi}\); Hans von Mangoldt proved this in 1905.

The explicit formula

Riemann counted prime powers with weights: \(J(x)=\sum_{n\ge1}\frac1n\pi(x^{1/n})\), where \(\pi(x)\) is the number of primes up to \(x\). Taking logarithms in the Euler product gives \(\log\zeta(s)=s\int_1^\infty J(x)x^{-s-1}\,dx\). Inverting this transform and using the product of \(\xi\) over its zeros, he obtained, for \(x \gt 1\) not a prime power,

with the sum taken over zeros in pairs \(\rho,\bar\rho\) in order of \(|\operatorname{Im}\rho|\). Möbius inversion then gives \(\pi(x)=\sum_n\frac{\mu(n)}{n}J(x^{1/n})\). Von Mangoldt proved the formula rigorously in 1895, in the equivalent form

The main term \(\operatorname{Li}(x)\) is Gauss’s guess. Each pair of zeros adds an oscillation of amplitude about \(x^{\operatorname{Re}\rho}\). Showing that no zero lies on \(\operatorname{Re}s=1\) gives the prime number theorem \(\pi(x)\sim x/\log x\), proved this way by Jacques Hadamard and Charles de la Vallée Poussin in 1896.

Riemann called it very probable and set it aside. Hilbert put it in his eighth problem in 1900. G. H. Hardy proved in 1914 that infinitely many zeros lie on the line. In 1932 Carl Ludwig Siegel found in Riemann’s papers an asymptotic formula Riemann had used to compute the first zeros by hand: the Riemann–Siegel formula, which the hero figure uses. Billions of zeros have since been checked numerically, all on the line. It is one of the Clay Millennium Prize Problems.

Hypergeometric functions and monodromy

For the general reader. Gauss had studied a power series that contains most of the special functions of physics as particular cases. Riemann characterized these functions without the series, by how they behave near three special points and how they change when you walk around those points. This idea, describing a function by its monodromy, became a central theme of twentieth-century mathematics.

The equation

Gauss’s series \(F(a,b;c;z)=\sum_{n\ge0}\frac{(a)_n(b)_n}{(c)_n\,n!}z^n\) solves

The singular points are \(0,1,\infty\), all regular singular. Near each, solutions behave like \((z-z_0)^\lambda\) times a power series, where the two exponents \(\lambda\) are: \(0,\,1-c\) at \(0\); \(0,\,c-a-b\) at \(1\); \(a,\,b\) at \(\infty\).

Möbius transformations move the three points to \(0,1,\infty\), and multiplying by \(z^\lambda(1-z)^\mu\) shifts exponents. Riemann used these two operations to derive Kummer’s 24 solutions of the hypergeometric equation and the quadratic transformations, almost without computation.

Monodromy

Continue a basis \((w_1,w_2)\) of solutions along a loop around a singular point. It returns as \((w_1,w_2)M\) for an invertible matrix \(M\). The map from loops to matrices is the monodromy representation of \(\pi_1(\mathbb P^1\setminus\{0,1,\infty\})\), a free group on two generators. Riemann’s key point was that the monodromy, together with the local exponents, determines the equation. In an unpublished fragment he asked whether every representation arises from some equation of this kind. Hilbert included the question as his twenty-first problem. The Riemann–Hilbert correspondence grew out of it.

Shock waves and Riemann invariants

For the general reader. A loud sound wave does not keep its shape. The louder parts travel faster, catch up with the quieter parts ahead, and the wave front steepens into a near-jump in pressure: a shock. In 1860 Riemann gave the first mathematical theory of this effect, and the methods he invented are still the starting point for gas dynamics.

Characteristics

For one-dimensional isentropic flow with density \(\rho\), velocity \(u\) and pressure \(p=p(\rho)\), the equations are \(\rho_t+(\rho u)_x=0\) and \(u_t+uu_x+p_x/\rho=0\). Let \(c=\sqrt{p'(\rho)}\) be the sound speed.

Derivation. Multiply the continuity equation by \(c/\rho\) and add or subtract it from the momentum equation, using \(p_x=c^2\rho_x\). The result is \(u_t\pm\frac c\rho\rho_t+(u\pm c)\big(u_x\pm\frac c\rho\rho_x\big)=0\), which is the claim.

So \(r\) and \(s\) are carried along two families of characteristic curves. Riemann used this to solve the initial value problem and showed that faster characteristics can overtake slower ones, which forces the solution to become discontinuous in finite time. He also posed what is now called the Riemann problem: the evolution of two constant states separated by a jump. Its solution, by shocks and rarefaction waves, is the building block of Godunov’s numerical method. Riemann assumed that the gas stays isentropic through the shock. That is wrong: energy conservation, not entropy, must hold across the jump. William Rankine (1870) and Pierre-Henri Hugoniot (1887) gave the correct jump conditions.

Professor, illness, Italy, 1857–1866

Gauss died in February 1855, and Dirichlet came from Berlin to take his chair. An attempt to obtain a professorship for Riemann failed. The 1857 paper on abelian functions made his reputation. In November 1857 he became extraordinary professor, with a salary. Dedekind, who had heard his lectures on abelian functions in 1855–56, became a close friend, and his later editions and memoir are the main source for Riemann’s life.

Dirichlet died on 5 May 1859. On 30 July 1859 Riemann succeeded him as full professor, holding the chair once held by Gauss. In August 1859 the Berlin Academy elected him a corresponding member, on a proposal by Ernst Kummer, Carl Wilhelm Borchardt and Karl Weierstrass. He visited Berlin, where he was received warmly, and sent the Academy the paper on prime numbers. The Göttingen Society of Sciences made him an ordinary member in December 1859. His students and listeners in these years included Gustav Roch and Friedrich Prym.

In June 1862 he married Elise Koch, a friend of his sister. That autumn he fell ill with pleurisy, followed by tuberculosis. He went south to recover and spent the winter of 1862–63 in Sicily, returned briefly, and went back south. His daughter Ida was born in Pisa in 1863. In Italy he spent time with Enrico Betti, who had visited Göttingen in 1858 and took up Riemann’s ideas on the connectivity of spaces. He lived in northern Italy from August 1864 to October 1865, and spent the following winter in Göttingen.

In 1866 honours arrived: foreign membership of the Berlin Academy in March, correspondence with the Paris Academy, and foreign membership of the Royal Society of London in June. In mid-June 1866, as the armies of Hanover and Prussia met near Göttingen, he left for Italy for the last time. He died at Selasca on Lake Maggiore on 20 July 1866, aged 39. He was buried in the cemetery of Biganzolo, where his gravestone survives in the cemetery wall. After his death, a housekeeper is said to have destroyed some of his unpublished papers.

Legacy

Riemann’s collected works, edited by Heinrich Weber with Dedekind’s help, appeared in 1876 and fill a single volume. Weber also edited his lectures on partial differential equations, which became a standard text of mathematical physics in later editions. Felix Klein spread Riemann’s geometric view of complex functions in lectures and a short book of 1882, against the more formal Berlin school of Weierstrass.

• Complex analysis and topology: Riemann surfaces, the mapping theorem and the idea of connectivity led through Betti and Henri Poincaré to algebraic topology, and through Weyl to the theory of manifolds.
• Algebraic geometry: genus, Riemann–Roch, theta functions and moduli are central to the theory of curves. Riemann–Roch was generalized by Friedrich Hirzebruch and by Grothendieck, and by Atiyah and Singer in the index theorem.
• Geometry and physics: Riemannian geometry is the language of general relativity.
• Number theory: the zeta function, the explicit formula and the Riemann hypothesis set the agenda of analytic number theory. The hypothesis is Hilbert’s eighth problem and a Clay Millennium Prize Problem.
• Analysis and PDE: the Riemann integral, the rearrangement theorem, the Riemann–Lebesgue lemma, Riemann invariants and the Riemann problem.

Sources and further reading

Biography

1. J. J. O’Connor and E. F. Robertson, Bernhard Riemann, MacTutor History of Mathematics.
2. Bernhard Riemann, Wikipedia.
3. H. Freudenthal, Riemann, Bernhard, Neue Deutsche Biographie 21 (2003).
4. Bernhard Riemann, Mathematics Genealogy Project.
5. R. Dedekind, Bernhard Riemann’s Lebenslauf, in Riemann, Gesammelte mathematische Werke, 1876.
6. D. Laugwitz, Bernhard Riemann 1826–1866: Turning Points in the Conception of Mathematics, Birkhäuser, 1999.

Mathematics

1. B. Riemann, Gesammelte mathematische Werke und wissenschaftlicher Nachlass, ed. H. Weber and R. Dedekind, Teubner, 1876. Texts and English translations by D. R. Wilkins.
2. B. Riemann, On the Hypotheses Which Lie at the Bases of Geometry, ed. J. Jost, Birkhäuser, 2016.
3. H. M. Edwards, Riemann’s Zeta Function, Academic Press, 1974 (reprinted by Dover), with a translation of the 1859 paper.
4. E. Bombieri, The Riemann Hypothesis, Clay Mathematics Institute problem description.
5. H. Weyl, Die Idee der Riemannschen Fläche, Teubner, 1913; English translation The Concept of a Riemann Surface.
6. J. Gray, Linear Differential Equations and Group Theory from Riemann to Poincaré, Birkhäuser, 2nd ed. 2000.
7. M. Spivak, A Comprehensive Introduction to Differential Geometry, vol. 2, Publish or Perish, with a commentary on the habilitation lecture.
8. R. Courant and K. O. Friedrichs, Supersonic Flow and Shock Waves, Interscience, 1948.