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The life of Bernhard Riemann

Timeline 1826Born on 17 September in Breselenz, in the Kingdom of Hanover, the son of a Lutheran pastor. 1846Enters the University of Göttingen to study theology, but soon moves to mathematics. 1847Goes to study in Berlin, with Dirichlet, Jacobi, Steiner and Eisenstein. 1851Doctorate in Göttingen, supervised by Gauss, with the thesis on complex functions. 1854On 10 June, gives his habilitation lecture on the foundations of geometry. 1857Publishes the theory of abelian functions and becomes an associate professor. 1859Succeeds Dirichlet in the chair at Göttingen and is elected to the Berlin Academy. Publishes the paper on prime numbers. 1862Marries Elise Koch. Falls ill with tuberculosis. 1866Dies on 20 July in Selasca, Italy, aged 39. The pastor’s son Georg Friedrich Bernhard Riemann was born on 17 September 1826 in Breselenz, a village in the Kingdom of Hanover, in northern Germany. His father, also named Friedrich Bernhard, was a Lutheran pastor who had fought in the wars against Napoleon. The family was poor, and Bernhard was the second of six children. It was his father who taught him until he was ten.

October 30, 2026 Read
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The Riemann integral

Newton and Leibniz invented integral calculus in the seventeenth century, but for a long time nobody worried about defining precisely what an integral is. For ordinary functions, the integral was simply the inverse of the derivative, or the area under the curve, and that was enough. Strange functions In the early nineteenth century the situation changed. Fourier showed that very irregular functions, with jumps and corners, could be written as sums of sines and cosines, and the coefficients of those sums were given by integrals. But what sense did it make to integrate a function full of jumps? Cauchy gave a first definition for continuous functions. Dirichlet, Riemann’s teacher, found in 1829 a function that could not be integrated at all: the one that equals 1 at the rationals and 0 at the irrationals.

October 31, 2026 Read
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The rearrangement theorem

Since school we learn that “the order of the terms does not change the sum”. For finite sums, it is true. For infinite sums, it is not, and the clearest example appears in the same habilitation work of 1854 in which Riemann defined the integral. A series that adds up to ln 2 The alternating harmonic series $$1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \frac{1}{5} - \cdots$$converges, and its sum is $\ln 2 \approx 0.6931$. But there is something fragile about it. If we separate the positive terms from the negative ones, each group on its own is infinite:

October 31, 2026 Read
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Riemann surfaces

Riemann’s doctoral thesis, of 1851, has a modest title: Foundations for a General Theory of Functions of a Complex Variable. Inside it are two ideas that changed mathematics: the conditions a complex function must satisfy in order to have a derivative, and a geometric way of dealing with functions that have several values. Functions that can be differentiated A complex function $f(z)$, with $z = x + iy$, can be written as $f = u + iv$, where $u$ and $v$ are real functions of $x$ and $y$. For $f$ to have a derivative in the complex sense, that is, for the limit $\frac{f(z + h) - f(z)}{h}$ to be the same whatever the direction in which $h$ goes to zero, the parts $u$ and $v$ must satisfy the Cauchy–Riemann equations:

October 31, 2026 Read
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The hypotheses on which geometry is founded

The question of the validity of the hypotheses of geometry in the infinitely small is bound up with the question of the ground of the metric relations of space. — Bernhard Riemann, On the Hypotheses which lie at the Bases of Geometry (1854) On 10 June 1854, Riemann presented to the Göttingen faculty the lecture On the Hypotheses which lie at the Bases of Geometry. He wrote it for an audience that included philosophers and historians, almost without formulas. The text was only published in 1868, after his death, and it changed geometry.

October 31, 2026 Read
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The zeta function

The function that bears Riemann’s name was studied earlier by Euler. For a real number $s$ greater than 1, the series $$\zeta(s) = 1 + \frac{1}{2^s} + \frac{1}{3^s} + \frac{1}{4^s} + \cdots = \sum_{n=1}^{\infty}\frac{1}{n^s}$$converges. Euler computed several of its values, such as $\zeta(2) = \frac{\pi^2}{6}$, the solution of the Basel problem. Euler’s product Euler’s deepest discovery, of 1737, was that the same function can be written as a product over the prime numbers:

October 31, 2026 Read
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Counting the prime numbers

The prime numbers seem to appear at random: 2, 3, 5, 7, 11, 13, then a gap until 17, 19, 23, another until 29. But seen from afar, they obey a rule. Call $\pi(x)$ the number of primes less than or equal to $x$. While still a teenager, Gauss noticed, looking at tables, that near a number $x$ the density of primes is about $\frac{1}{\ln x}$, and proposed the estimate $$\pi(x) \approx \operatorname{li}(x) = \int_0^x \frac{dt}{\ln t},$$the logarithmic integral. Nobody knew how to prove that it was good, nor how to explain the deviations.

October 31, 2026 Read
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The Riemann hypothesis

In the middle of the 1859 paper on prime numbers, Riemann writes a sentence almost in passing. Speaking of a function linked to zeta, whose real roots correspond to the zeros of zeta on the critical line, he says: It is very probable that all the roots are real. One would of course wish for a rigorous proof of this; but I have, after some fleeting vain attempts, provisionally put aside the search for one, since it did not seem necessary for the next objective of my investigation. — Bernhard Riemann, On the Number of Primes Less Than a Given Magnitude (1859)

October 31, 2026 Read
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The legacy of Bernhard Riemann

Riemann is the man of brilliant intuition. — Felix Klein, Lectures on the Development of Mathematics in the 19th Century (1926) Riemann’s collected works, edited by Dedekind and Weber in 1876, fit into a volume of a little over five hundred pages, including the posthumous works and the notes. Few mathematicians left so little text and so many consequences. What bears his name The Riemann integral and Riemann sums, the definition of the integral in calculus courses. Riemann surfaces and the Riemann sphere, the complex plane with a point at infinity, closed up like a sphere. The Cauchy–Riemann equations and the Riemann mapping theorem. Riemannian geometry, Riemannian manifolds, the metric and the Riemann curvature tensor, the language of general relativity. The Riemann–Roch theorem, which counts the functions that can be defined on a Riemann surface, completed by his student Gustav Roch in 1865. The Riemann zeta function and the Riemann hypothesis. The Riemann rearrangement theorem, on conditionally convergent series. A new style More than the results, Riemann changed the style of mathematics. Before him, a function was above all a formula, and mathematicians worked by manipulating formulas. Riemann thought in concepts: a complex function defined by its properties, not by its expression; a surface defined without being drawn anywhere; a space defined by its rule of measurement. Twentieth-century mathematicians, from Hilbert to Weyl, saw in him the founder of the “conceptual” mathematics that became dominant.

October 31, 2026 Read
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Image credits

Historical images and photographs All come from Wikimedia Commons. We use the files or the thumbnails provided by Commons, reduced to at most 800 pixels wide, with no other editing. Portrait of Bernhard Riemann from 1863: source page · public domain. The young Riemann, photograph from the 1850s: source page · public domain. Stone in memory of Riemann beside the church of Breselenz, photo by Felina Katarina Schulz: source page · CC BY-SA 4.0. Plaque on a house in Göttingen where Riemann lived, photo by Manecke: source page · CC BY-SA 3.0. Riemann surface of the square root, figure by Leonid 2 (PNG version rendered by Commons from the SVG): source page · CC BY-SA 3.0. Riemann’s gravestone in the cemetery of Biganzolo, photo by g.rondena: source page · public domain. Figures made for the blog The cover and the four interactive figures (Riemann sums, the rearrangement of the alternating harmonic series, the count of primes with the function R(x), and the zeta function on the critical line) are original illustrations, with text in each of the site’s languages.

October 31, 2026 Read