RiemannSiegelTheta
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.
Examples
open allclose allBasic Examples (6)
Plot over a subset of the reals:
Plot over a subset of the complexes:
Series expansion at the origin:
Series expansion at Infinity:
Scope (26)
Numerical Evaluation (4)
Specific Values (2)
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:
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)
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)
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:
Text
Wolfram Research (1991), RiemannSiegelTheta, Wolfram Language function, https://reference.wolfram.com/language/ref/RiemannSiegelTheta.html.
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