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RamanujanTauTheta

RamanujanTauTheta[t]

gives the Ramanujan tau theta function .

Details

Mathematical function, suitable for both symbolic and numerical manipulation.
for real .
arises in the study of the Ramanujan L-function on the critical line. It is closely related to the number of zeros of for .
Apart from a sign, gives the phase of the Ramanujan L-function .
is an analytic function of except for branch cuts on the imaginary axis running from to .
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 (6)

Evaluate numerically:

Plot over a subset of the reals:

Plot over a subset of the complexes:

Series expansion at the origin:

Series expansion at Infinity:

Series expansion at a singular point:

Scope (27)

Numerical Evaluation (7)

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 RamanujanTauTheta function using MatrixFunction:

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 (10)

RamanujanTauTheta is defined for all real values:

Complex domain:

Function range of RamanujanTauTheta:

RamanujanTauTheta threads over lists:

RamanujanTauTheta is an analytic function of x:

RamanujanTauTheta is neither non-increasing nor non-decreasing:

RamanujanTauTheta is not injective:

RamanujanTauTheta is surjective:

RamanujanTauTheta is neither non-negative nor non-positive:

RamanujanTauTheta has no singularities or discontinuities:

RamanujanTauTheta is neither convex nor concave:

Differentiation (2)

First derivative with respect to :

Higher derivatives with respect to :

Plot the higher derivatives with respect to :

Series Expansions (4)

Find the Taylor expansion using Series:

Plots of the first three approximations around :

Find the series expansion at Infinity:

Find the series expansion for an arbitrary symbolic direction :

Taylor expansion at a generic point:

Applications (4)

Contour plot of the absolute value of RamanujanTauTheta:

The first 10 Gram points of RamanujanTauL:

Plot of RamanujanTauZ and Gram points:

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

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

Properties & Relations (3)

RamanujanTauTheta is related to LogGamma:

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

RamanujanTauZ can be expressed in terms of RamanujanTauTheta and RamanujanTauL:

Possible Issues (1)

Machine-number inputs can give high-precision results:

Neat Examples (2)

Density plot of the argument:

Riemann surface of RamanujanTauTheta:

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