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The extended Riemann hypothesis for abelian extension of the rationals is equivalent to the generalized Riemann hypothesis. The Riemann hypothesis can also be extended to the L-functions of Hecke characters of number fields. The grand Riemann hypothesis extends it to all automorphic zeta functions, such as Mellin transforms of Hecke eigenforms.
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 . The equation is also known as the Papperitz equation. [1]
The Riemann hypothesis is one of the most important conjectures in mathematics.It is a statement about the zeros of the Riemann zeta function.Various geometrical and arithmetical objects can be described by so-called global L-functions, which are formally similar to the Riemann zeta-function.
The Riemann Hypothesis. ... The definition: A real number is algebraic if it’s the root of some polynomial with integer coefficients. For example, x²-6 is a polynomial with integer coefficients ...
The Riemann zeta function is an example of an L-function, and some important conjectures involving L-functions are the Riemann hypothesis and its generalizations. The theory of L -functions has become a very substantial, and still largely conjectural , part of contemporary analytic number theory .
In mathematics, the grand Riemann hypothesis is a generalisation of the Riemann hypothesis and generalized Riemann hypothesis. It states that the nontrivial zeros of all automorphic L -functions lie on the critical line 1 2 + i t {\displaystyle {\frac {1}{2}}+it} with t {\displaystyle t} a real number variable and i {\displaystyle i} the ...
The generalized Riemann hypothesis is the conjecture that all the non-trivial zeros lie on the critical line Re(s) = 1/2. [ 9 ] Up to the possible existence of a Siegel zero , zero-free regions including and beyond the line Re( s ) = 1 similar to that of the Riemann zeta function are known to exist for all Dirichlet L -functions: for example ...
A Riemann problem, named after Bernhard Riemann, is a specific initial value problem composed of a conservation equation together with piecewise constant initial data which has a single discontinuity in the domain of interest.