RiemannSiegelZ

RiemannSiegelZ[t]

gives the RiemannSiegel function .

Details

  • Mathematical function, suitable for both symbolic and numerical manipulation.
  • , where is the RiemannSiegel theta function, and is the Riemann zeta function.
  • for real .
  • is an analytic function of except for branch cuts on the imaginary axis running from to .
  • For certain special arguments, RiemannSiegelZ automatically evaluates to exact values.
  • RiemannSiegelZ can be evaluated to arbitrary numerical precision.
  • RiemannSiegelZ automatically threads over lists.

Examples

open allclose all

Basic Examples  (6)

Evaluate numerically:

Find a numerical root:

Plot over a subset of the reals:

Plot over a subset of the complexes:

Series expansion at the origin:

Series expansion at a singular point:

Scope  (19)

Numerical Evaluation  (4)

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 the first positive maximum of RiemannSiegelZ[x]:

Visualization  (3)

Plot the RiemannSiegelZ:

Plot the real part of the RiemannSiegelZ function:

Plot the imaginary part of the RiemannSiegelZ function:

Plot the real part of the RiemannSiegelZ function:

Plot the imaginary part of the RiemannSiegelZ function:

Function Properties  (5)

RiemannSiegelZ is defined for all real values:

Complex domain:

RiemannSiegelZ is defined through the identity:

RiemannSiegelZ threads elementwise over lists:

TraditionalForm formatting:

Differentiation  (3)

First derivative with respect to :

Evaluate derivatives numerically:

First and second derivatives with respect to :

Plot the first and second derivatives with respect to :

Series Expansions  (2)

Find the Taylor expansion using Series:

Plots of the first three approximations around :

Taylor expansion at a generic point:

Applications  (5)

Plot real and imaginary parts over the complex plane:

View on the branch cut along the imaginary axis:

Find a zero of ReimannSiegelZ using FindRoot:

Or using ZetaZero:

Find several zeros:

Plot curves of vanishing real and imaginary parts of RiemannSiegelZ:

A version of the Riemann hypothesis requires the limit of as to vanish:

Plot double logarithmically the value of the integral:

Calculate a "signal power" of the Riemann zeta function along the critical line:

Plot the difference to the asymptotic value:

Properties & Relations  (2)

Relation to the Riemann zeta function:

Numerically find a root of a transcendental equation:

Possible Issues  (2)

A larger setting for $MaxExtraPrecision can be needed:

Machine-number inputs can give highprecision results:

Neat Examples  (3)

Recurrence plot of RiemannSiegelZ:

Play RiemannSiegelZ as a sound:

Animate RiemannSiegelZ:

Introduced in 1991
 (2.0)