Bernhard Riemann
September 17, 1826 — July 20, 1866
Bernhard Riemann was a German mathematician who, in a career cut short at 39 by tuberculosis, produced work of such originality and depth that it reshaped geometry, analysis, and number theory, provided Albert Einstein with the mathematical language of general relativity, and left behind in the Riemann hypothesis one of the most famous unsolved problems in the history of mathematics.
Early Life and Education
Born on September 17, 1826 in Breselenz, in the Kingdom of Hanover (now Germany), Georg Friedrich Bernhard Riemann was the son of a Lutheran pastor and one of six children. He showed exceptional mathematical talent from childhood; a story holds that at age 14, borrowing a 900-page text on number theory from his teacher, he returned it six days later having mastered it. He studied at the University of Göttingen under Carl Friedrich Gauss, the greatest mathematician of the age, and later at the University of Berlin. Gauss reportedly saw in Riemann's doctoral thesis — on the theory of complex functions — "a gloriously fertile originality." Riemann's life was shadowed by poverty, illness, and the burden of supporting his family, but the mathematics he produced in spite of these circumstances remains extraordinary in its depth and originality.
Riemannian Geometry and the Shape of Space
In 1854, Riemann delivered his habilitation lecture — required for professorship — entitled "On the Hypotheses that Lie at the Foundation of Geometry." It was one of the most consequential mathematical talks ever given. In it, Riemann extended geometry beyond Euclid's flat plane and beyond the curved surfaces of ordinary 3D space, conceiving a mathematics of n-dimensional curved spaces of arbitrary shape. He introduced what became known as the Riemannian metric: a way of measuring distances on any curved surface or space. Sixty years later, Albert Einstein used Riemannian geometry as the mathematical language of his general theory of relativity, in which gravity is understood as the curvature of four-dimensional spacetime. Without Riemann, there is no general relativity in the form Einstein gave it. Gauss, who heard the 1854 lecture, was reported to have been moved beyond his usual reserve by what he heard.
Did You Know?
The Riemann hypothesis, stated in an 1859 paper, proposes that all non-trivial zeros of the Riemann zeta function lie on the critical line ½ + it in the complex plane. Despite being one of the most intensively studied problems in mathematics for 165 years, it remains unproved. It is one of the seven Millennium Prize Problems for which the Clay Mathematics Institute offers a prize of $1 million. Mathematicians have verified computationally that the first 10 trillion zeros lie on the critical line, but a general proof remains elusive.
The Riemann Hypothesis and Legacy
In 1859, Riemann published "On the Number of Primes Less Than a Given Magnitude," an eight-page paper that is arguably the most influential paper in the history of mathematics. In it he introduced the Riemann zeta function and stated the hypothesis about its zeros that has occupied number theorists ever since. He also introduced the Riemann integral, which provided the rigorous foundation for calculus that the subject had always lacked. He became a professor at Göttingen in 1859, but his health — always fragile — deteriorated rapidly. He died of tuberculosis on July 20, 1866 in Selasca, Italy, on the shore of Lake Maggiore, during a rest cure, at the age of 39. His housekeeper, following the custom of the time, burned many of his unpublished papers after his death — a loss that mathematicians still mourn.