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RiemannSiegelTheta

RiemannSiegelTheta[t]

gives the Riemann–Siegel function .

Details

Mathematical function, suitable for both symbolic and numerical manipulation.
for real .
arises in the study of the Riemann zeta function on the critical line. It is closely related to the number of zeros of for .
is an analytic function of except for branch cuts on the imaginary axis running from to .
For certain special arguments, RiemannSiegelTheta automatically evaluates to exact values.
RiemannSiegelTheta can be evaluated to arbitrary numerical precision.
RiemannSiegelTheta automatically threads over lists.
RiemannSiegelTheta can be used with Interval and CenteredInterval objects. »

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 (28)

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 worst-case guaranteed intervals using Interval and CenteredInterval objects:

Or compute average-case statistical intervals using Around:

Compute the elementwise values of an array:

Or compute the matrix RiemannSiegelTheta function using MatrixFunction:

Specific Values (2)

Values at zero:

Find the positive minimum of RiemannSiegelTheta[x]:

Visualization (2)

Plot the RiemannSiegelTheta:

Plot the real part of the RiemannSiegelTheta function:

Plot the imaginary part of the RiemannSiegelTheta function:

Function Properties (11)

RiemannSiegelTheta is defined for all real values:

Complex domain:

Function range of RiemannSiegelTheta:

RiemannSiegelTheta threads elementwise over lists:

RiemannSiegelTheta is an analytic function of x:

RiemannSiegelTheta is non-increasing in a specific range:

RiemannSiegelTheta is not injective:

RiemannSiegelTheta is surjective:

RiemannSiegelTheta is neither non-negative nor non-positive:

RiemannSiegelTheta has no singularities or discontinuities:

RiemannSiegelTheta is neither convex nor concave:

TraditionalForm formatting:

Differentiation (3)

First derivative with respect to :

Higher derivatives with respect to :

Plot the higher derivatives with respect to :

Formula for the derivative 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:

Generalizations & Extensions (2)

Series expansion at the origin:

Series expansion at a branch point:

Applications (3)

Plot real and imaginary parts over the complex plane:

Show interlacing of the roots of Sin[RiemannSiegelTheta[t]] and RiemannSiegelZ[t]:

Compute Gram points:

Show good Gram points, where RiemannSiegelZ changes sign for consecutive points:

Show a bad Gram point:

Properties & Relations (3)

RiemannSiegelTheta is related to LogGamma:

RiemannSiegelZ can be expressed in terms of RiemannSiegelTheta and Zeta:

Numerically find a root of a transcendental equation:

Possible Issues (2)

A larger setting for $MaxExtraPrecision might be needed:

Machine-number inputs can give high‐precision results:

Neat Examples (1)

Riemann surface of RiemannSiegelTheta:

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