RamanujanTauTheta

RamanujanTauTheta[t]

gives the Ramanujan tau theta function .

Details

  • Mathematical function, suitable for both symbolic and numerical manipulation.
  • Apart from a sign, gives the phase of the Ramanujan L-function .
  • For certain special arguments, RamanujanTauTheta automatically evaluates to exact values.
  • RamanujanTauTheta can be evaluated to arbitrary numerical precision.
  • RamanujanTauTheta automatically threads over lists.

Examples

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Basic Examples  (3)

Evaluate numerically:

Plot over a subset of the reals:

Plot over a subset of the complexes:

Scope  (12)

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:

Specific Values  (2)

Value at zero:

Find positive minimum of RamanujanTauTheta[x]:

Visualization  (2)

Plot the RamanujanTauTheta:

Plot the real part of RamanujanTauTheta function:

Plot the imaginary part of RamanujanTauTheta function:

Function Properties  (3)

RamanujanTauTheta is defined for all real values:

Complex domain:

Function range of RamanujanTauTheta:

RamanujanTauTheta threads over lists:

Applications  (2)

Contour plot of the absolute value of RamanujanTauTheta:

The first 10 Gram points of RamanujanTauL:

Plot of RamanujanTauZ and Gram points:

Properties & Relations  (1)

On the critical line, RamanujanTauTheta gives the phase of RamanujanTauL up to a sign:

RamanujanTauL can be expressed in terms of RamanujanTauTheta:

Possible Issues  (1)

Machine-number inputs can give high-precision results:

Neat Examples  (1)

Density plot of the argument:

Introduced in 2007
 (6.0)