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Riemann made some famous contributions to modern analytic number theory. In a single short paper, the only one he published on the subject of number theory, he investigated the zeta function that now bears his name, establishing its importance for understanding the distribution of prime numbers.
Riemann hypothesis, in number theory, hypothesis by German mathematician Bernhard Riemann concerning the location of solutions to the Riemann zeta function, which is connected to the prime number theorem and has important implications for the distribution of prime numbers.
Riemann's thesis studied the theory of complex variables and, in particular, what we now call Riemann surfaces. It therefore introduced topological methods into complex function theory.
Bernhard Riemann (1826 - 1866) - Biography - MacTutor History of ...
The Riemann hypothesis is a claim about a mathematical function so gnarly that for most numbers fed as its inputs, no one knows its exact output.
Bernhard Riemann was a pioneering mathematician whose contributions to the fields of analysis, number theory, and differential geometry have had a lasting impact on the field.
The subject founded by this work is Riemannian geometry. Riemann found the correct way to extend into n dimensions the differential geometry of surfaces, which Gauss himself proved in his theorema egregium. The fundamental objects are called the Riemannian metric and the Riemann curvature tensor.
Bernhard Riemann made profound, far-sighted discoveries with lasting consequences for mathematics and our understanding of space, gravity, and time. Riemannian geometry completely reformed the field of geometry and became the mathematical foundation of Einstein's general theory of relativity.
