RamanujanTau

RamanujanTau[n]

gives the Ramanujan function .

Details

  • Integer mathematical function.
  • gives the coefficient of in the series expansion of .
  • RamanujanTau automatically threads over lists.

Examples

open allclose all

Basic Examples  (2)

The first 10 values of RamanujanTau:

Plot over a subset of the reals:

Scope  (12)

Numerical Evaluation  (3)

Evaluate numerically:

Evaluate efficiently for large values of the argument:

Compute the elementwise values of an array:

Or compute the matrix RamanujanTau function using MatrixFunction:

Specific Values  (2)

Values at fixed points:

Value at zero:

Visualization  (3)

Plot the RamanujanTau function:

Plot the contours of the RamanujanTau function:

Plot the RamanujanTau function in three dimensions:

Function Properties  (4)

RamanujanTau is only defined for integer inputs:

RamanujanTau threads over lists:

RamanujanTauL is everywhere singular:

Traditional form:

Applications  (7)

Logarithmic plot of RamanujanTau:

The first prime value of RamanujanTau:

The first 20,000 values are nonzero, satisfying Lehmer's conjecture [more info]:

Plot of at primes :

The modular discriminant:

Relation with DedekindEta:

The summatory -function [more info]:

The modular discriminant:

Relation with DedekindEta:

Properties & Relations  (7)

The first 10 values of RamanujanTau using Product:

RamanujanTau is multiplicative for coprime integers:

For prime :

Congruence relations:

Representation of an integer as the sum of 24 squares:

RamanujanTauL is the Dirichlet -function associated with RamanujanTau:

FindSequenceFunction can recognize the RamanujanTau sequence:

Possible Issues  (1)

Large prime numbers can take a long time:

Neat Examples  (3)

Successive differences of RamanujanTau modulo 3:

A representation of zero in terms of RamanujanTau:

Find digit counts for RamanujanTau[10^12]:

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

Text

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

CMS

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

APA

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

BibTeX

@misc{reference.wolfram_2024_ramanujantau, author="Wolfram Research", title="{RamanujanTau}", year="2007", howpublished="\url{https://reference.wolfram.com/language/ref/RamanujanTau.html}", note=[Accessed: 21-December-2024 ]}

BibLaTeX

@online{reference.wolfram_2024_ramanujantau, organization={Wolfram Research}, title={RamanujanTau}, year={2007}, url={https://reference.wolfram.com/language/ref/RamanujanTau.html}, note=[Accessed: 21-December-2024 ]}