RiemannSiegelZ

RiemannSiegelZ[t]

gives the RiemannSiegel function TemplateBox[{t}, RiemannSiegelZ].

Details

  • Mathematical function, suitable for both symbolic and numerical manipulation.
  • TemplateBox[{t}, RiemannSiegelZ]=ⅇ^(ⅈ TemplateBox[{t}, RiemannSiegelTheta])TemplateBox[{{{1, /, 2}, +, {ⅈ,  , t}}}, Zeta], where is the RiemannSiegel theta function, and is the Riemann zeta function.
  • |TemplateBox[{t}, RiemannSiegelZ]|=|TemplateBox[{{{1, /, 2}, +, {ⅈ,  , t}}}, Zeta]| for real .
  • TemplateBox[{t}, RiemannSiegelZ] 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.
  • RiemannSiegelZ can be used with Interval and CenteredInterval objects. »

Examples

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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  (26)

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:

RiemannSiegelZ can be used with Interval and CenteredInterval objects:

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  (11)

RiemannSiegelZ is defined for all real values:

Complex domain:

RiemannSiegelZ is defined through the identity:

RiemannSiegelZ threads elementwise over lists:

RiemannSiegelZ is an analytic function of x:

RiemannSiegelZ is neither non-increasing nor non-decreeing:

RiemannSiegelZ is not injective:

RiemannSiegelZ is neither non-negative nor non-positive:

RiemannSiegelZ does not have singularity or discontinuity:

RiemannSiegelZ is neither convex nor concave:

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  (6)

Plot real and imaginary parts over the complex plane:

View on the branch cut along the imaginary axis:

Find a zero of RiemannSiegelZ 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 from the asymptotic value:

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

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:

Wolfram Research (1991), RiemannSiegelZ, Wolfram Language function, https://reference.wolfram.com/language/ref/RiemannSiegelZ.html (updated 2023).

Text

Wolfram Research (1991), RiemannSiegelZ, Wolfram Language function, https://reference.wolfram.com/language/ref/RiemannSiegelZ.html (updated 2023).

CMS

Wolfram Language. 1991. "RiemannSiegelZ." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2023. https://reference.wolfram.com/language/ref/RiemannSiegelZ.html.

APA

Wolfram Language. (1991). RiemannSiegelZ. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/RiemannSiegelZ.html

BibTeX

@misc{reference.wolfram_2023_riemannsiegelz, author="Wolfram Research", title="{RiemannSiegelZ}", year="2023", howpublished="\url{https://reference.wolfram.com/language/ref/RiemannSiegelZ.html}", note=[Accessed: 16-April-2024 ]}

BibLaTeX

@online{reference.wolfram_2023_riemannsiegelz, organization={Wolfram Research}, title={RiemannSiegelZ}, year={2023}, url={https://reference.wolfram.com/language/ref/RiemannSiegelZ.html}, note=[Accessed: 16-April-2024 ]}