# RiemannSiegelZ

gives the RiemannSiegel function .

# Details • Mathematical function, suitable for both symbolic and numerical manipulation.
• , where is the RiemannSiegel theta function, and is the Riemann zeta function.
• for real .
• is an analytic function of except for branch cuts on the imaginary axis running from to .
• For certain special arguments, RiemannSiegelZ automatically evaluates to exact values.
• RiemannSiegelZ can be evaluated to arbitrary numerical precision.
• RiemannSiegelZ automatically threads over lists.

# Examples

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## Basic Examples(6)

Evaluate numerically:

Find a numerical root:

Plot over a subset of the reals:

Plot over a subset of the complexes:

Series expansion at the origin:

Series expansion at a singular point:

## Scope(25)

### Numerical Evaluation(4)

Evaluate numerically:

Evaluate to high precision:

The precision of the output tracks the precision of the input:

Complex number inputs:

Evaluate efficiently at high precision:

### Specific Values(2)

Value at zero:

Find the first positive maximum of :

### Visualization(3)

Plot the RiemannSiegelZ:

Plot the real part of the RiemannSiegelZ function:

Plot the imaginary part of the RiemannSiegelZ function:

Plot the real part of the RiemannSiegelZ function:

Plot the imaginary part of the RiemannSiegelZ function:

### Function Properties(11)

RiemannSiegelZ is defined for all real values:

Complex domain:

RiemannSiegelZ is defined through the identity:

RiemannSiegelZ threads elementwise over lists:

RiemannSiegelZ is an analytic function of x:

RiemannSiegelZ is neither non-increasing nor non-decreeing:

RiemannSiegelZ is not injective:

RiemannSiegelZ is neither non-negative nor non-positive:

RiemannSiegelZ does not have singularity or discontinuity:

RiemannSiegelZ is neither convex nor concave:

### Differentiation(3)

First derivative with respect to :

Evaluate derivatives numerically:

First and second derivatives with respect to :

Plot the first and second derivatives with respect to :

### Series Expansions(2)

Find the Taylor expansion using Series:

Plots of the first three approximations around :

Taylor expansion at a generic point:

## Applications(5)

Plot real and imaginary parts over the complex plane:

View on the branch cut along the imaginary axis:

Find a zero of ReimannSiegelZ using FindRoot:

Or using ZetaZero:

Find several zeros:

Plot curves of vanishing real and imaginary parts of RiemannSiegelZ:

A version of the Riemann hypothesis requires the limit of as to vanish:

Plot double logarithmically the value of the integral:

Calculate a "signal power" of the Riemann zeta function along the critical line:

Plot the difference to the asymptotic value:

## Properties & Relations(2)

Relation to the Riemann zeta function:

Numerically find a root of a transcendental equation:

## Possible Issues(2)

A larger setting for \$MaxExtraPrecision can be needed: Machine-number inputs can give highprecision results:

## Neat Examples(3)

Recurrence plot of RiemannSiegelZ:

Play RiemannSiegelZ as a sound:

Animate RiemannSiegelZ: