Bernhard Riemann · aiwiki.page

Bernhard Riemann, fully Georg Friedrich Bernhard Riemann (17 September 1826–20 July 1866), was a German mathematician whose research reshaped analysis, geometry, and number theory. He developed geometric methods for complex functions, established an influential definition of integration, and introduced the foundations of geometry on curved spaces of arbitrary dimension. His investigations of the zeta function connected complex analysis with prime-number distribution and produced the conjecture now called the Riemann hypothesis. Much of his career was spent at the University of Göttingen. (mathshistory.st-andrews.ac.uk)

Life and academic career

Riemann was born in Breselenz in the Kingdom of Hanover, now part of Germany, into the family of a Lutheran pastor. He attended schools in Hanover and Lüneburg before entering Göttingen in 1846, initially to study theology. With his father’s permission, he changed to mathematics. In 1847 he moved to Berlin, where his teachers included Peter Gustav Lejeune Dirichlet and Carl Gustav Jacob Jacobi; he returned to Göttingen in 1849. (mathshistory.st-andrews.ac.uk)

His doctoral dissertation, completed in 1851 under Carl Friedrich Gauss, concerned functions of a complex variable. He qualified to lecture through his habilitation in 1854, became an extraordinary professor in 1857, and succeeded Dirichlet in Göttingen’s mathematics chair in 1859. In 1862 he married Elise Koch; they had one daughter. Illness led to extended stays in Italy, where he died at Selasca on Lake Maggiore, aged thirty-nine. (mathshistory.st-andrews.ac.uk)

Complex functions and Riemann surfaces

Riemann’s dissertation, Foundations for a General Theory of Functions of a Complex Variable, treated complex analysis through the interaction of analytical and geometric ideas. For a function written as $f(z)=u(x,y)+iv(x,y)$ , complex differentiability imposes the Cauchy–Riemann equations, relating the partial derivatives of its real and imaginary parts. His approach emphasized these structural conditions and the geometry of mappings, rather than relying exclusively on formulas and series. (maths.tcd.ie)

A central construction was the Riemann surface. Expressions such as square roots and logarithms can take multiple values when considered over the complex plane. By organizing their branches on an appropriate surface, one can treat them as single-valued functions. This made the connectivity of the underlying surface an essential ingredient in understanding analytic functions, linking function theory with topology. His 1857 paper on Abelian functions extended this program to more general integration and inversion problems. (maths.tcd.ie)

Some existence arguments depended on the Dirichlet principle, a variational method. Karl Weierstrass subsequently identified a gap in its unrestricted use: a lower bound for a functional does not by itself guarantee that a minimizing function exists. Later work, including that of David Hilbert, supplied rigorous foundations under suitable conditions. (mathshistory.st-andrews.ac.uk)

Integration and trigonometric series

Riemann’s habilitation dissertation investigated representation by Fourier series and clarified which functions possess a definite integral. Submitted in 1854, it was published posthumously by Richard Dedekind in 1867. The work combined a historical examination of trigonometric series with new investigations, including a definition accommodating classes of discontinuous functions. (maths.tcd.ie)

The Riemann integral is obtained by dividing an interval into subintervals, choosing sample points, and forming Riemann sums:

\sum_i f(\xi_i)(x_i-x_{i-1}).

The integral exists when these sums approach the same value as the largest subinterval length tends to zero, independently of the partitions and sample points. This separates the question of integrability from the stronger assumption of continuity. Riemann’s treatment also examined what can be inferred about a function from its representation by a trigonometric series, reversing the usual question of finding conditions that guarantee such a representation. (maths.tcd.ie)

Geometry of curved spaces

On 10 June 1854, Riemann delivered his habilitation lecture, On the Hypotheses Which Lie at the Foundations of Geometry. He distinguished an $n$ -dimensional manifold from the additional structure needed to measure lengths on it. The same underlying continuum could admit different measurement rules; ordinary Euclidean geometry was therefore a particular case rather than the only conceivable geometry. (aps.org)

In modern notation, the structure he investigated is expressed by a metric tensor:

ds^2=\sum_{i,j}g_{ij}(x)\,dx^i\,dx^j.

For the positive-definite case, this specifies infinitesimal lengths and thereby lengths of curves. Riemann generalized intrinsic curvature from surfaces to higher dimensions and studied geodesics, the local counterparts of straight lines. These ideas became foundational for differential geometry and Riemannian geometry. (maths.tcd.ie)

The lecture also distinguished mathematical possibilities from the empirical question of which geometry describes physical space. Its framework later contributed to Albert Einstein’s general relativity, in which gravity is described through curved spacetime. This was a later physical application, not a theory formulated by Riemann himself. (maths.tcd.ie)

Zeta function and prime numbers

In November 1859, Riemann published On the Number of Primes Less Than a Given Magnitude. He investigated the Riemann zeta function,

\zeta(s)=\sum_{n=1}^{\infty}n^{-s}, \qquad \operatorname{Re}(s)>1,

using a complex variable, extending the function beyond the initial region of convergence and establishing a functional equation. His analysis connected its zeros with formulas for counting prime numbers. The paper distinguished proved arguments from assertions whose complete justification remained to be supplied. (maths.tcd.ie)

The Riemann hypothesis states that every nontrivial zero of this function has real part $1/2$ . It concerns the fluctuations of prime counts around their average distribution, rather than merely that average itself. The conjecture remains unresolved and is listed by the Clay Mathematics Institute among its Millennium Prize Problems. (claymath.org)

Other research and publication

Riemann also investigated finite-amplitude air waves, electrical phenomena, and the motion of a fluid ellipsoid. His published and posthumous papers include studies of theta functions, surfaces of least area, and electrodynamics. Dedekind and Heinrich Weber helped preserve and publish his work; several foundational texts became widely available only after his death. (maths.tcd.ie)