Dirichlet L-function oor Frans
Dirichlet L-function
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Examples of primitive functions include the Riemann zeta function and Dirichlet L-functions of primitive Dirichlet characters.
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Dirichlet L-functions are generalizations of the Riemann zeta-function and appear prominently in the generalized Riemann hypothesis.
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It is a particular Dirichlet L-function, the L-function for the alternating character of period four.
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The generalized Riemann hypothesis (for Dirichlet L-functions) was probably formulated for the first time by Adolf Piltz in 1884.
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This is known for one-dimensional representations, the L-functions being then associated to Hecke characters — and in particular for Dirichlet L-functions.
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Hecke had earlier related Dirichlet L-functions with automorphic forms (holomorphic functions on the upper half plane of C that satisfy certain functional equations).
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The insight of Langlands was to find the proper generalization of Dirichlet L-functions, which would allow the formulation of Artin's statement in this more general setting.
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It is often said to have begun with Peter Gustav Lejeune Dirichlet's 1837 introduction of Dirichlet L-functions to give the first proof of Dirichlet's theorem on arithmetic progressions.
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Global L-functions can be associated to elliptic curves, number fields (in which case they are called Dedekind zeta-functions), Maass forms, and Dirichlet characters (in which case they are called Dirichlet L-functions).
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The best results on the structure of the Selberg class are due to Kaczorowski and Perelli, who show that the Dirichlet L-functions (including the Riemann zeta-function) are the only examples with degree less than 2.
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When the Riemann hypothesis is formulated for Dedekind zeta-functions, it is known as the extended Riemann hypothesis (ERH) and when it is formulated for Dirichlet L-functions, it is known as the generalized Riemann hypothesis (GRH).
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In mathematics, more specifically in the field of analytic number theory, a Siegel zero, named after Carl Ludwig Siegel, is a type of potential counterexample to the generalized Riemann hypothesis, on the zeroes of Dirichlet L-function.
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By using a theorem by Carl Ludwig Siegel providing an upper bound for the real zeros (see Siegel zero) of Dirichlet L-functions formed with real non-principal characters, Walfisz obtained the Siegel-Walfisz theorem, from which the prime number theorem for arithmetic progressions can be deduced.
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In proving the theorem, he introduced the Dirichlet characters and L-functions.
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EUDML | A note on the real zeros of Dirichlet's L-functions
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We distinguish at the outset between the L-series, an infinite series representation (for example the Dirichlet series for the Riemann zeta function), and the L-function, the function in the complex plane that is its analytic continuation.
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Jutila's proof requires significant adaptations but we succeed to extend these results to a large class of Dirichlet L-functions.
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A conventional starting point for analytic number theory is Dirichlet's theorem on arithmetic progressions (1837), whose proof introduced L-functions and involved some asymptotic analysis and a limiting process on a real variable.
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Roughly speaking, proving that some set of primes has a non-zero Dirichlet density usually involves showing that certain L-functions do not vanish at the point s = 1, while showing that they have a natural density involves showing that the L-functions have no zeros on the line Re(s) = 1.
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In it, broad generalisations of the Riemann zeta function and the L-series for a Dirichlet character are constructed, and their general properties, in most cases still out of reach of proof, are set out in a systematic way.
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These characters and their associated L-functions are now understood to be strictly related to complex multiplication, as the Dirichlet characters are to cyclotomic fields.
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The members of the class are Dirichlet series which obey four axioms that seem to capture the essential properties satisfied by most functions that are commonly called L-functions or zeta functions.
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Math., 113 (1998), 237-346] to the functional equation of L-functions of Dirichlet type associated with prehomogeneous vector spaces.
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► In the early 20th century, Erich Hecke attempted to find a further generalization of the Dirichlet L-series and the Dedekind zeta function.
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Here, we shall reexamine the harmonic (as well as hilbertian and distributional) meaning of the functional equations with the simplest shape, the one applying to the Riemann zeta function and to the Dirichlet L-series (many of our considerations have a more general range.)
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