# 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**

### 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.

### Ramanujan's master theorem

Ramanujan's master theorem)

### 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.

### List of mathematical functions

**elementary functionsList of functionsMathematical functions**