Georg Friedrich Bernhard Riemann (1826–1866) lived only 39 years and left behind a body of work that reshaped three separate fields of mathematics. His 1859 paper on the zeta function posed a question — the Riemann Hypothesis — that remains unanswered after more than 160 years and is widely considered the most important open problem in mathematics.
Life
Riemann was born in the Hanover kingdom, the son of a Lutheran pastor. The family was poor, and Riemann’s health was fragile from the start. He studied at Göttingen, where he attended lectures by Gauss, and then at Berlin, where he was influenced by Dirichlet, Jacobi, and Eisenstein.
He returned to Göttingen, where Gauss supervised his doctoral thesis in 1851. His 1854 Habilitation lecture, titled Über die Hypothesen welche der Geometrie zu Grunde liegen (“On the Hypotheses which Lie at the Foundations of Geometry”), stunned Gauss — who was famously hard to impress.
Riemann succeeded Dirichlet as professor at Göttingen in 1859. But his tuberculosis had already taken hold. He spent increasing time recuperating in Italy and died there in 1866 at age 39.
Contributions
Riemannian geometry
Riemann’s 1854 Habilitation lecture introduced what we now call Riemannian geometry — the mathematical framework for studying spaces that are locally Euclidean but globally curved. He generalized Gauss’s work on surfaces to arbitrary dimensions and introduced the metric tensor as the central object of the theory.
Sixty years later, Albert Einstein would build general relativity on exactly this mathematical foundation. Without Riemann’s geometry, Einstein could not have formulated his theory. “Had I known of Riemann’s work earlier, I would have reached my final results much sooner,” Einstein said.
Complex analysis and Riemann surfaces
Riemann’s doctoral thesis (1851) introduced Riemann surfaces: a way of handling multi-valued complex functions by thinking of them as single-valued on a more complicated surface. This transformed complex analysis, revealing deep connections between complex functions, topology, and algebraic geometry.
The Riemann integral
The Riemann integral, the first rigorous definition of integration for a broad class of functions, is from Riemann’s Habilitation thesis (1854). It dominated integration theory until Lebesgue’s more general theory in 1904.
The zeta function and the Riemann Hypothesis
In 1859 Riemann published his only paper on number theory, Über die Anzahl der Primzahlen unter einer gegebenen Größe (“On the Number of Prime Numbers Less than a Given Magnitude”). In just eight pages he:
- Extended the zeta function to the complex plane
- Connected its zeros to the distribution of primes through an explicit formula
- Conjectured that all non-trivial zeros lie on the “critical line” with real part
This Riemann Hypothesis has become the most famous unsolved problem in mathematics. The paper set the agenda for analytic number theory for the next century and a half.
Legacy
Riemann died far too young. His published output was modest — a few dozen papers — but every one opened a new field or transformed an existing one. His students and successors (Dedekind, Klein, Hilbert) carried his ideas forward and made them the backbone of 19th- and 20th-century mathematics.
It is fair to say that modern geometry, modern complex analysis, and modern number theory all run through Göttingen around 1860. The Riemann Hypothesis — his most famous conjecture — remains one of mathematics’ great unfinished sentences.
Known for
- Riemann Hypothesis
- Riemannian geometry
- Riemann surfaces
- Riemann zeta function
Frequently asked
What is Riemann's most important contribution?
Impossible to choose. Riemannian geometry became the framework for general relativity. Riemann surfaces transformed complex analysis. His 1859 paper on the zeta function set the agenda for analytic number theory, and the Riemann Hypothesis within it remains the most famous unsolved problem in mathematics.
Why did he die so young?
Tuberculosis. Riemann suffered from fragile health for much of his life and died at 39 while recuperating in Italy. Given what he achieved in such a short life, his early death is one of mathematics' great lost continuations.