# RiemannSiegelTheta

gives the RiemannSiegel function .

# Details • Mathematical function, suitable for both symbolic and numerical manipulation.
• for real .
• arises in the study of the Riemann zeta function on the critical line. It is closely related to the number of zeros of for .
• is an analytic function of except for branch cuts on the imaginary axis running from to .
• For certain special arguments, RiemannSiegelTheta automatically evaluates to exact values.
• RiemannSiegelTheta can be evaluated to arbitrary numerical precision.
• RiemannSiegelTheta automatically threads over lists.

# Examples

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## Basic Examples(6)

Evaluate numerically:

Plot over a subset of the reals:

Plot over a subset of the complexes:

Series expansion at the origin:

Series expansion at Infinity:

Series expansion at a singular point:

## Scope(26)

### 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)

Values at zero:

Find the positive minimum of :

### Visualization(2)

Plot the RiemannSiegelTheta:

Plot the real part of the RiemannSiegelTheta function:

Plot the imaginary part of the RiemannSiegelTheta function:

### Function Properties(11)

RiemannSiegelTheta is defined for all real values:

Complex domain:

Function range of RiemannSiegelTheta:

RiemannSiegelTheta is an analytic function of x:

RiemannSiegelTheta is non-increasing in a specific range:

RiemannSiegelTheta is not injective:

RiemannSiegelTheta is surjective:

RiemannSiegelTheta is neither non-negative nor non-positive:

RiemannSiegelTheta has no singularities or discontinuities:

RiemannSiegelTheta is neither convex nor concave:

### Differentiation(3)

First derivative with respect to :

Higher derivatives with respect to :

Plot the higher derivatives with respect to :

Formula for the  derivative with respect to :

### Series Expansions(4)

Find the Taylor expansion using Series:

Plots of the first three approximations around :

Find the series expansion at Infinity:

Find the series expansion for an arbitrary symbolic direction :

Taylor expansion at a generic point:

## Generalizations & Extensions(2)

Series expansion at the origin:

Series expansion at a branch point:

## Applications(2)

Plot real and imaginary parts over the complex plane:

Show interlacing of the roots of Sin[RiemannSiegelTheta[t]] and :

## Properties & Relations(2)

RiemannSiegelTheta is related to LogGamma:

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(1)

Riemann surface of RiemannSiegelTheta: