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

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Basic Examples  (2)

The first 10 values of RamanujanTau:

Plot over a subset of the reals:

Scope  (10)

Numerical Evaluation  (2)

Evaluate numerically:

Evaluate efficiently for large values of the argument:

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  (3)

Real domain of RamanujanTau:

Complex domain:

RamanujanTau threads over lists:

Traditional form:

Applications  (6)

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]:

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]:

Introduced in 2007
 (6.0)