# 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.
• RiemannSiegelTheta can be used with Interval and CenteredInterval objects. »

# 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(27)

### Numerical Evaluation(5)

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:

RiemannSiegelTheta can be used with Interval and CenteredInterval objects:

### 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 threads elementwise over lists:

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:

TraditionalForm formatting:

### 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(3)

Plot real and imaginary parts over the complex plane:

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

Compute Gram points:

Show good Gram points, where RiemannSiegelZ changes sign for consecutive points:

Show a bad Gram point:

## Properties & Relations(3)

RiemannSiegelTheta is related to LogGamma:

RiemannSiegelZ can be expressed in terms of RiemannSiegelTheta and Zeta:

Numerically find a root of a transcendental equation:

## Possible Issues(2)

A larger setting for \$MaxExtraPrecision might be needed:

Machine-number inputs can give highprecision results:

## Neat Examples(1)

Riemann surface of RiemannSiegelTheta:

Wolfram Research (1991), RiemannSiegelTheta, Wolfram Language function, https://reference.wolfram.com/language/ref/RiemannSiegelTheta.html (updated 2023).

#### Text

Wolfram Research (1991), RiemannSiegelTheta, Wolfram Language function, https://reference.wolfram.com/language/ref/RiemannSiegelTheta.html (updated 2023).

#### CMS

Wolfram Language. 1991. "RiemannSiegelTheta." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2023. https://reference.wolfram.com/language/ref/RiemannSiegelTheta.html.

#### APA

Wolfram Language. (1991). RiemannSiegelTheta. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/RiemannSiegelTheta.html

#### BibTeX

@misc{reference.wolfram_2024_riemannsiegeltheta, author="Wolfram Research", title="{RiemannSiegelTheta}", year="2023", howpublished="\url{https://reference.wolfram.com/language/ref/RiemannSiegelTheta.html}", note=[Accessed: 12-July-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_riemannsiegeltheta, organization={Wolfram Research}, title={RiemannSiegelTheta}, year={2023}, url={https://reference.wolfram.com/language/ref/RiemannSiegelTheta.html}, note=[Accessed: 12-July-2024 ]}