# RamanujanTauZ

gives the Ramanujan tau Z-function .

# Details

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

# Examples

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## Basic Examples(3)

Evaluate numerically:

Plot over a subset of the reals:

Series expansion at the origin:

## Scope(20)

### 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(3)

Value at zero:

RamanujanTauZ evaluates to exact values for certain special arguments:

Find the first positive maximum of :

### Visualization(2)

Plot the RamanujanTauZ:

Plot the real part of the RamanujanTauZ function:

Plot the imaginary part of the RamanujanTauZ function:

### Function Properties(9)

RamanujanTauZ is defined for all real values:

Complex domain:

Approximate function range of RamanujanTauZ:

RamanujanTauZ is an analytic function of x:

RamanujanTauZ is neither non-increasing nor non-decreasing:

RamanujanTauZ is not injective:

RamanujanTauZ is neither non-negative nor non-positive:

RamanujanTauZ has no singularities or discontinuities:

RamanujanTauZ is neither convex nor concave:

### Series Expansions(2)

Find the Taylor expansion using Series:

Taylor expansion at a generic point:

## Applications(4)

Plot of the absolute value of RamanujanTauZ:

Find a numerical root:

On the critical line, RamanujanTauL splits:

Show interlacing of the roots of Sin[RamanujanTauTheta[t]] and RamanujanTauZ[t]]:

## Properties & Relations(3)

Relation to the Ramanujan tau L-function:

On the critical line, RamanujanTauZ is the modulus of RamanujanTauL up to a sign:

RamanujanTauZ can be expressed in terms of RamanujanTauL and RamanujanTauTheta:

Evaluate derivatives numerically:

Wolfram Research (2007), RamanujanTauZ, Wolfram Language function, https://reference.wolfram.com/language/ref/RamanujanTauZ.html.

#### Text

Wolfram Research (2007), RamanujanTauZ, Wolfram Language function, https://reference.wolfram.com/language/ref/RamanujanTauZ.html.

#### CMS

Wolfram Language. 2007. "RamanujanTauZ." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/RamanujanTauZ.html.

#### APA

Wolfram Language. (2007). RamanujanTauZ. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/RamanujanTauZ.html

#### BibTeX

@misc{reference.wolfram_2023_ramanujantauz, author="Wolfram Research", title="{RamanujanTauZ}", year="2007", howpublished="\url{https://reference.wolfram.com/language/ref/RamanujanTauZ.html}", note=[Accessed: 25-September-2023 ]}

#### BibLaTeX

@online{reference.wolfram_2023_ramanujantauz, organization={Wolfram Research}, title={RamanujanTauZ}, year={2007}, url={https://reference.wolfram.com/language/ref/RamanujanTauZ.html}, note=[Accessed: 25-September-2023 ]}