RiemannSiegelTheta

RiemannSiegelTheta[t]

gives the Riemann–Siegel function .

Details

  • Mathematical function, suitable for both symbolic and numerical manipulation.
  • theta(t)=Im(log Gamma (1/4+ it/2))-t/2log pi 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.

Examples

open allclose all

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  (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)

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  (4)

RiemannSiegelTheta is defined for all real values:

Complex domain:

Function range of RiemannSiegelTheta:

RiemannSiegelTheta threads elementwise over lists:

TraditionalForm formatting:

Differentiation  (3)

First derivative with respect to :

Higher derivatives with respect to :

Plot the higher derivatives with respect to :

Formula for the ^(th) 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  (2)

Plot real and imaginary parts over the complex plane:

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

Properties & Relations  (2)

RiemannSiegelTheta is related to LogGamma:

Numerically find a root of a transcendental equation:

Possible Issues  (2)

A larger setting for $MaxExtraPrecision can be needed:

Machine-number inputs can give high‐precision results:

Neat Examples  (1)

Riemann surface of RiemannSiegelTheta:

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

Text

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

BibTeX

@misc{reference.wolfram_2020_riemannsiegeltheta, author="Wolfram Research", title="{RiemannSiegelTheta}", year="1991", howpublished="\url{https://reference.wolfram.com/language/ref/RiemannSiegelTheta.html}", note=[Accessed: 19-April-2021 ]}

BibLaTeX

@online{reference.wolfram_2020_riemannsiegeltheta, organization={Wolfram Research}, title={RiemannSiegelTheta}, year={1991}, url={https://reference.wolfram.com/language/ref/RiemannSiegelTheta.html}, note=[Accessed: 19-April-2021 ]}

CMS

Wolfram Language. 1991. "RiemannSiegelTheta." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/RiemannSiegelTheta.html.

APA

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