RamanujanTauL

RamanujanTauL[s]

gives the Ramanujan tau Dirichlet L-function .

Details

  • Mathematical function, suitable for both symbolic and numerical manipulation.
  • For Re(s)>6, is given by the Dirichlet series , where is the Ramanujan function.
  • RamanujanTauL can be evaluated to arbitrary numerical precision.
  • RamanujanTauL automatically threads over lists.

Examples

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Basic Examples  (2)

Evaluate numerically:

Plot over a subset of the reals:

Scope  (20)

Numerical Evaluation  (6)

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:

Compute average-case statistical intervals using Around:

Compute the elementwise values of an array:

Or compute the matrix RamanujanTauL function using MatrixFunction:

Specific Values  (2)

Value at zero:

Find a value of x for which RamanujanTauL[x]=0.8:

Visualization  (2)

Plot the RamanujanTauL:

Plot the real part of RamanujanTauL function:

Plot the imaginary part of RamanujanTauL function:

Function Properties  (10)

RamanujanTauL is defined for all real values:

Complex domain:

Bounds on the function range of RamanujanTauL:

RamanujanTauL threads over lists:

RamanujanTauL is an analytic function of x:

RamanujanTauL is neither non-increasing nor non-decreasing:

RamanujanTauL is not injective:

RamanujanTauL is surjective:

RamanujanTauL is neither non-negative nor non-positive:

RamanujanTauL has no singularities or discontinuities:

RamanujanTauL is neither convex nor concave:

Applications  (5)

Plot on the critical line:

Find a zero of RamanujanTauL:

The number of zeros on the critical strip from 0 to :

Plot of the real part:

Define and visualize the Ramanujan function:

Properties & Relations  (5)

Functional equation:

Approximate RamanujanTauL using an Euler product formula:

On the critical line, RamanujanTauL can be split into RamanujanTauTheta and RamanujanTauZ:

Evaluate derivatives numerically:

RamanujanTauZ can be expressed using RamanujanTauL:

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

Text

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

CMS

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

APA

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

BibTeX

@misc{reference.wolfram_2025_ramanujantaul, author="Wolfram Research", title="{RamanujanTauL}", year="2007", howpublished="\url{https://reference.wolfram.com/language/ref/RamanujanTauL.html}", note=[Accessed: 13-May-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_ramanujantaul, organization={Wolfram Research}, title={RamanujanTauL}, year={2007}, url={https://reference.wolfram.com/language/ref/RamanujanTauL.html}, note=[Accessed: 13-May-2025 ]}