Riesz function

Riesz criterionRiesz-type formula
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.wikipedia
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Riemann hypothesis

Critical line theoremcritical line1st
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
The Riesz criterion was given by, to the effect that the bound

Marcel Riesz

RieszRiesz MarcelRiesz, Marcel
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
He also introduced the Riesz function Riesz(x), and showed that the Riemann hypothesis is equivalent to the bound Riesz(x) = O(x 1⁄4 + ε ) as x → ∞, for any ε > 0.

Bernoulli number

Bernoulli numbersgeneralized Bernoulli numberAn identity
denotes the rising factorial power in the notation of D. E. Knuth and the number B n are the Bernoulli number.
is the Riesz function

Mathematics

mathematicalmathmathematician
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

Entire function

entireHadamard productorder
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

Falling and rising factorials

Pochhammer symbolfalling factorialrising factorial
denotes the rising factorial power in the notation of D. E. Knuth and the number B n are the Bernoulli number.

Donald Knuth

Donald E. KnuthKnuthDon Knuth
denotes the rising factorial power in the notation of D. E. Knuth and the number B n are the Bernoulli number.

Big O notation

Obig-O notationlittle-o notation
for any exponent e larger than 1/2, where this is big O notation; taking values both positive and negative.

Riemann zeta function

zeta functionRiemann zeta-functionRiemann's zeta function
The Riesz function is related to the Riemann zeta function via its Mellin transform.

Mellin transform

Cahen-Mellin integralCahen–Mellin integralMellin
The Riesz function is related to the Riemann zeta function via its Mellin transform.

Borel summation

Borel sumBorel resummationBorel transform
J. garcia (see references) gave the integral representation of f(x) using Borel resummation as

Taylor series

Taylor expansionMaclaurin seriesTaylor polynomial
The Maclaurin series coefficients of F increase in absolute value until they reach their maximum at the 40th term of -1.753.

Möbius function

Moebius functionMobius functionμ(''n'')
:Here μ is the Möbius mu function, and the rearrangement of terms is justified by absolute convergence.

Edward Charles Titchmarsh

E. C. TitchmarshEdward TitchmarshTitchmarsh
*Titchmarsh, E. C., The Theory of the Riemann Zeta Function, second revised (Heath-Brown) edition, Oxford University Press, 1986, [Section 14.32]

Ali Abkar

Abkar, Ali
He is best known for his works on Riesz-type formula, for some works in Nonlinear Functional Analysis, as well as for contributions to The Theory of Bergman spaces.