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.
  • 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  (27)

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:

RiemannSiegelTheta can be used with Interval and CenteredInterval objects:

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 ^(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  (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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