In mathematics, the Riesz function is an entire function defined by Marcel Riesz in connection with the Riemann hypothesis, by means of the power series
If we set we may define it in terms of the coefficients of the Laurent series development of the hyperbolic (or equivalently, the ordinary) cotangent around zero. If
then F may be defined as
The values of ζ(2k) approach one for increasing k, and comparing the series for the Riesz function with that for shows that it defines an entire function. Alternatively, F may be defined as
denotes the rising factorial power in the notation of D. E. Knuth and the number Bn are the Bernoulli number. The series is one of alternating terms and the function quickly tends to minus infinity for increasingly negative values of x. Positive values of x are more interesting and delicate.
In mathematics, the Rieszfunction is an entire function defined by Marcel Riesz in connection with the Riemann hypothesis, by means of the power series...
function Complete Fermi–Dirac integral, an alternate form of the polylogarithm. Dilogarithm Incomplete Fermi–Dirac integral Kummer's functionRiesz function...
Marcel Riesz (Hungarian: Riesz Marcell [ˈriːs ˈmɒrt͡sɛll]; 16 November 1886 – 4 September 1969) was a Hungarian mathematician, known for work on summation...
mathematics, the Riesz potential is a potential named after its discoverer, the Hungarian mathematician Marcel Riesz. In a sense, the Riesz potential defines...
In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order. This concept...
integral on the space of all compactly supported continuous functions φ which, by the Riesz representation theorem, can be represented as the Lebesgue...
The Riesz representation theorem, sometimes called the Riesz–Fréchet representation theorem after Frigyes Riesz and Maurice René Fréchet, establishes...
terms, and are only readily visible when y < 10−5. The Riesz mean of the von Mangoldt function is given by ∑ n ≤ λ ( 1 − n λ ) δ Λ ( n ) = − 1 2 π i ∫...
Bourbaki group (Bourbaki 1987) they were first introduced by Frigyes Riesz (Riesz 1910). Lp spaces form an important class of Banach spaces in functional...
convolution of one function with another function having a singularity at the origin. Specifically, the Riesz transforms of a complex-valued function ƒ on Rd are...
circle. This result, the Herglotz-Riesz representation theorem, was proved independently by Gustav Herglotz and Frigyes Riesz in 1911. It can be used to give...
a Riesz space, lattice-ordered vector space or vector lattice is a partially ordered vector space where the order structure is a lattice. Riesz spaces...
"dubious lament". At the 1908 international congress of mathematics F. Riesz introduced an alternate way defining limits and continuity in concept called...
Riesz mean should not be confused with the Bochner–Riesz mean or the Strong–Riesz mean. Given a series { s n } {\displaystyle \{s_{n}\}} , the Riesz mean...
measure in D {\displaystyle D} . This is called the Riesz representation theorem. Subharmonic functions are of a particular importance in complex analysis...
for all x {\displaystyle x} in H . {\displaystyle H.} It follows from the Riesz representation theorem that any symmetric (defined as a ( x , y ) = a (...
operators". The classical results about interpolation of operators are the Riesz–Thorin theorem and the Marcinkiewicz theorem. There are also many other...
certain spaces of holomorphic functions on the unit disk or upper half plane. They were introduced by Frigyes Riesz (Riesz 1923), who named them after G...
of ΔSn−1. In particular, an application of the spectral theorem to the Riesz potential Δ S n − 1 − 1 {\displaystyle \Delta _{S^{n-1}}^{-1}} gives another...
complete and careful presentation of the theory. Good presentation of the Riesz extension theorems. However, there is a minor flaw (in the first edition)...
(depending on ε) such that |R(x)| < Cεxε as x → ∞. Here R(x) is the Rieszfunction R ( x ) = 2 ∑ k = 1 ∞ k k ¯ x k ( 2 π ) 2 k ( B 2 k 2 k ) = 2 ∑ k =...
mathematics, the Riesz rearrangement inequality, sometimes called Riesz–Sobolev inequality, states that any three non-negative functions f : R n → R + {\displaystyle...
Radon measure by the Riesz–Markov–Kakutani representation theorem. If the function space of locally integrable functions, i.e. functions belonging to L loc...
Banach spaces originally grew out of the study of function spaces by Hilbert, Fréchet, and Riesz earlier in the century. Banach spaces play a central...
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