In mathematics, the Hirzebruch–Riemann–Roch theorem, named after Friedrich Hirzebruch, Bernhard Riemann, and Gustav Roch, is Hirzebruch's 1954 result contributing to the Riemann–Roch problem for complex algebraic varieties of all dimensions.
In mathematics, Riemann's differential equation, named after Bernhard Riemann, is a generalization of the hypergeometric differential equation, allowing the regular singular points to occur anywhere on the Riemann sphere, rather than merely at 0, 1, and ∞.
Bernhard Riemann | Riemann zeta function | Riemann surface | Riemann hypothesis | Riemann sphere | Riemann solver | Riemann-Roch theorem | Hugo Riemann | Riemann Xi function | Riemann's function | Riemann's existence theorem | Riemann–Roch theorem | Riemann manifold | Riemann integral | Riemann | Generalized Riemann Hypothesis | Fritz Riemann |
In the early 20th century G. H. Hardy and Littlewood proved many results about the zeta function in an attempt to prove the Riemann Hypothesis.
Numerical tests of the Riemann hypothesis use this technique to get an upper bound for the number of zeros of Eugene Trubowitz
with Joel Feldman, Horst Knörrer: Riemann Surfaces of Infinite Genus, AMS (American Mathematical Society) 2003
(This result is due to Adrien Douady for 0 characteristic and has its origins in Riemann's existence theorem. For a field of arbitrary characteristic it is due to David Harbater and Florian Pop, and was also proved later by Dan Haran and Moshe Jarden.)
He studied from 1924 to 1929 at the universities of Graz and Göttingen and received his doctor's degree in 1929 under Richard Courant at Georg August University of Göttingen with the thesis about "Verallgemeinerung der Riemannschen Integrationsmethode auf Differentialgleichungen n-ter Ordnung in zwei Veränderlichen" ("Generalization of Riemann's integration method on differential equations of n-th order in two variables").
The Hilbert transform is named after David Hilbert, who first introduced the operator in order to solve a special case of the Riemann–Hilbert problem for holomorphic functions.
In 1857, Riemann introduced the concept of Riemann surfaces as part of a study of the process of analytic continuation; Riemann surfaces are now recognized as one-dimensional complex manifolds.
With Joel Feldman, Eugene Trubowitz: Riemann Surfaces of Infinite Genus, AMS (American Mathematical Society) 2003
For example, consider a subset X of {0,1} specified by the following rule: 0 belongs to X if and only if the Riemann hypothesis is true, and 1 belongs to X if and only if the Riemann hypothesis is false.
Katja Hannchen Leni Riemann (born 1 November 1963 in Weyhe-Kirchweyhe, Germany) is a German actress.
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Born as the daughter of two teachers, Katja Riemann spent her childhood in Weyhe, near Bremen.
In 1997 Deshouillers, Effinger, te Riele, and Zinoviev showed that the generalized Riemann hypothesis implies that every odd number greater than 5 is the sum of three primes.
Riemann invariants are constant along the characteristic curves of the partial differential equations where they obtain the name invariant.
The theorem for compact Riemann surfaces can be deduced from the algebraic version using Chow's theorem and the GAGA principle: in fact, every compact Riemann surface is defined by algebraic equations in some complex projective space.
According to lectures published by Heinrich Martin Weber in 1901, the real and imaginary components of the equation
Its development was rapid in the years after 1950, when it was realised that sheaf cohomology was connected with more classical methods applied to the Riemann-Roch theorem, the analysis of a linear system of divisors in algebraic geometry, several complex variables, and Hodge theory.
Using the superconnection formalism of Quillen, they obtained a refinement of the Riemann–Roch formula, which links together the Thom classes in K-theory and cohomology, as an equality on the level of differential forms.
He has contributed to one of the discoveries concerning the Riemann zeta function, namely, that the Riemann zeros appear to display the same statistics as those which are believed to be present in energy levels of quantum chaotic systems and described by Random Matrix Theory.